Av(1234, 1324, 2341, 3412)
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Generating Function
\(\displaystyle \frac{2 x^{7}-3 x^{6}+16 x^{5}-31 x^{4}+33 x^{3}-21 x^{2}+7 x -1}{\left(2 x -1\right) \left(x -1\right)^{6}}\)
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Counting Sequence
1, 1, 2, 6, 20, 63, 180, 468, 1126, 2553, 5542, 11666, 24048, 48899, 98592, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(2 x -1\right) \left(x -1\right)^{6} F \! \left(x \right)-2 x^{7}+3 x^{6}-16 x^{5}+31 x^{4}-33 x^{3}+21 x^{2}-7 x +1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 63\)
\(\displaystyle a \! \left(6\right) = 180\)
\(\displaystyle a \! \left(7\right) = 468\)
\(\displaystyle a \! \left(n +1\right) = 2 a \! \left(n \right)-\frac{\left(n -1\right) \left(n^{4}-19 n^{3}+66 n^{2}-154 n +60\right)}{60}, \quad n \geq 8\)
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Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0 \\ -6+6 \,2^{n}+\frac{n^{5}}{60}-\frac{n^{4}}{4}+\frac{7 n^{3}}{12}-\frac{13 n^{2}}{4}-\frac{21 n}{10} & \text{otherwise} \end{array}\right.\)
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This specification was found using the strategy pack "Point And Row And Col Placements Req Corrob Expand Verified" and has 87 rules.

Found on January 21, 2022.

Finding the specification took 85 seconds.

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\(\begin{align*} F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{1}\! \left(x \right) &= 1\\ F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{4}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{4}\! \left(x \right) &= x\\ F_{5}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{4}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{13}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{10}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{17}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{60}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{11}\! \left(x \right) F_{22}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{36}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{31}\! \left(x \right)+F_{35}\! \left(x \right)\\ F_{30}\! \left(x \right) &= 0\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{34}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{4}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{29}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{11}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{37}\! \left(x \right)+F_{54}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{38}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{44}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{41}\! \left(x \right)+F_{42}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{11}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{4}\! \left(x \right) F_{43}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{40}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{45}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{46}\! \left(x \right)+F_{50}\! \left(x \right)+F_{56}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{4}\! \left(x \right) F_{47}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)+F_{49}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{40}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{45}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{4}\! \left(x \right) F_{51}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{52}\! \left(x \right)+F_{54}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{4}\! \left(x \right) F_{53}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{51}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{4}\! \left(x \right) F_{55}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{51}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{4}\! \left(x \right) F_{57}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)+F_{59}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{35}\! \left(x \right)\\ F_{59}\! \left(x \right) &= 2 F_{30}\! \left(x \right)+F_{50}\! \left(x \right)+F_{56}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{61}\! \left(x \right)+F_{81}\! \left(x \right)\\ F_{61}\! \left(x \right) &= -F_{79}\! \left(x \right)+F_{62}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{5}\! \left(x \right)+F_{63}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{4}\! \left(x \right) F_{64}\! \left(x \right)\\ F_{64}\! \left(x \right) &= \frac{F_{65}\! \left(x \right)}{F_{4}\! \left(x \right)}\\ F_{65}\! \left(x \right) &= F_{66}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{67}\! \left(x \right)+F_{71}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{68}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{69}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{4}\! \left(x \right) F_{70}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{68}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{72}\! \left(x \right)+F_{76}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{73}\! \left(x \right)+F_{75}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{4}\! \left(x \right) F_{74}\! \left(x \right)\\ F_{74}\! \left(x \right) &= F_{7}\! \left(x \right)+F_{72}\! \left(x \right)\\ F_{75}\! \left(x \right) &= F_{4}\! \left(x \right) F_{66}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{77}\! \left(x \right)\\ F_{77}\! \left(x \right) &= F_{4}\! \left(x \right) F_{78}\! \left(x \right)\\ F_{78}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{76}\! \left(x \right)\\ F_{79}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{80}\! \left(x \right)\\ F_{80}\! \left(x \right) &= F_{10}\! \left(x \right) F_{11}\! \left(x \right)\\ F_{81}\! \left(x \right) &= F_{82}\! \left(x \right)\\ F_{82}\! \left(x \right) &= F_{10}\! \left(x \right) F_{11}\! \left(x \right) F_{4}\! \left(x \right) F_{83}\! \left(x \right)\\ F_{83}\! \left(x \right) &= F_{84}\! \left(x \right)+F_{85}\! \left(x \right)\\ F_{84}\! \left(x \right) &= F_{10} \left(x \right)^{2}\\ F_{85}\! \left(x \right) &= F_{11}\! \left(x \right) F_{86}\! \left(x \right)\\ F_{86}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{11}\! \left(x \right)\\ \end{align*}\)

This specification was found using the strategy pack "Pattern Point Placements Req Corrob Expand Verified" and has 124 rules.

Found on January 21, 2022.

Finding the specification took 139 seconds.

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\(\begin{align*} F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{1}\! \left(x \right) &= 1\\ F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{4}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{8}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{8}\! \left(x \right) &= x\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{11}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{14}\! \left(x \right)+F_{18}\! \left(x \right)\\ F_{13}\! \left(x \right) &= 0\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{17}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{8}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{12}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{19}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{92}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right) F_{27}\! \left(x \right)\\ F_{26}\! \left(x \right) &= x^{2}\\ F_{27}\! \left(x \right) &= -F_{90}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= \frac{F_{29}\! \left(x \right)}{F_{26}\! \left(x \right)}\\ F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{47}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{41}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{35}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{36}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{37}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{39}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{38}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{42}\! \left(x \right)+F_{43}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{19}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{44}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{46}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{8}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{41}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)+F_{76}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{49}\! \left(x \right)+F_{66}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{50}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{51}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{48}\! \left(x \right)+F_{52}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{53}\! \left(x \right)+F_{57}\! \left(x \right)+F_{61}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{54}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)+F_{56}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{41}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{52}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{58}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{59}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{60}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{58}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{62}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{63}\! \left(x \right)+F_{64}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{18}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{65}\! \left(x \right)\\ F_{65}\! \left(x \right) &= 2 F_{13}\! \left(x \right)+F_{57}\! \left(x \right)+F_{61}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{67}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{68}\! \left(x \right)+F_{69}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{58}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{70}\! \left(x \right)+F_{73}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{66}\! \left(x \right)+F_{71}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{72}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{70}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{74}\! \left(x \right)\\ F_{74}\! \left(x \right) &= F_{75}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{75}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{73}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{57}\! \left(x \right)+F_{77}\! \left(x \right)+F_{80}\! \left(x \right)\\ F_{77}\! \left(x \right) &= F_{78}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{78}\! \left(x \right) &= F_{55}\! \left(x \right)+F_{79}\! \left(x \right)\\ F_{79}\! \left(x \right) &= F_{76}\! \left(x \right)\\ F_{80}\! \left(x \right) &= F_{8}\! \left(x \right) F_{81}\! \left(x \right)\\ F_{81}\! \left(x \right) &= F_{82}\! \left(x \right)+F_{86}\! \left(x \right)\\ F_{82}\! \left(x \right) &= F_{83}\! \left(x \right)\\ F_{83}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{18}\! \left(x \right)+F_{84}\! \left(x \right)\\ F_{84}\! \left(x \right) &= F_{8}\! \left(x \right) F_{85}\! \left(x \right)\\ F_{85}\! \left(x \right) &= F_{8}\! \left(x \right)+F_{83}\! \left(x \right)\\ F_{86}\! \left(x \right) &= F_{87}\! \left(x \right)\\ F_{87}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{57}\! \left(x \right)+F_{80}\! \left(x \right)+F_{88}\! \left(x \right)\\ F_{88}\! \left(x \right) &= F_{8}\! \left(x \right) F_{89}\! \left(x \right)\\ F_{89}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{87}\! \left(x \right)\\ F_{90}\! \left(x \right) &= F_{91}\! \left(x \right)\\ F_{91}\! \left(x \right) &= F_{21} \left(x \right)^{4} F_{8}\! \left(x \right)\\ F_{92}\! \left(x \right) &= F_{8}\! \left(x \right) F_{93}\! \left(x \right)\\ F_{93}\! \left(x \right) &= F_{94}\! \left(x \right)+F_{98}\! \left(x \right)\\ F_{94}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{95}\! \left(x \right)\\ F_{95}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{14}\! \left(x \right)+F_{96}\! \left(x \right)\\ F_{96}\! \left(x \right) &= F_{8}\! \left(x \right) F_{97}\! \left(x \right)\\ F_{97}\! \left(x \right) &= F_{6}\! \left(x \right)+F_{95}\! \left(x \right)\\ F_{98}\! \left(x \right) &= F_{99}\! \left(x \right)\\ F_{99}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{104}\! \left(x \right)\\ F_{100}\! \left(x \right) &= F_{101}\! \left(x \right) F_{19}\! \left(x \right) F_{21}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{101}\! \left(x \right) &= F_{102}\! \left(x \right)+F_{103}\! \left(x \right)\\ F_{102}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{103}\! \left(x \right) &= F_{19}\! \left(x \right) F_{21}\! \left(x \right)\\ F_{104}\! \left(x \right) &= F_{105}\! \left(x \right)\\ F_{105}\! \left(x \right) &= F_{21} \left(x \right)^{2} F_{8} \left(x \right)^{2} F_{106}\! \left(x \right)\\ F_{106}\! \left(x \right) &= F_{107}\! \left(x \right)+F_{114}\! \left(x \right)\\ F_{107}\! \left(x \right) &= F_{101}\! \left(x \right) F_{108}\! \left(x \right)\\ F_{108}\! \left(x \right) &= F_{109}\! \left(x \right)+F_{21}\! \left(x \right)\\ F_{109}\! \left(x \right) &= F_{110}\! \left(x \right)+F_{19}\! \left(x \right)\\ F_{110}\! \left(x \right) &= F_{111}\! \left(x \right)+F_{113}\! \left(x \right)+F_{13}\! \left(x \right)\\ F_{111}\! \left(x \right) &= F_{112}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{112}\! \left(x \right) &= F_{110}\! \left(x \right)+F_{19}\! \left(x \right)\\ F_{113}\! \left(x \right) &= F_{109}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{114}\! \left(x \right) &= F_{115}\! \left(x \right)+F_{120}\! \left(x \right)\\ F_{115}\! \left(x \right) &= F_{116}\! \left(x \right)+F_{19}\! \left(x \right)\\ F_{116}\! \left(x \right) &= F_{117}\! \left(x \right)+F_{119}\! \left(x \right)+F_{13}\! \left(x \right)\\ F_{117}\! \left(x \right) &= F_{118}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{118}\! \left(x \right) &= F_{116}\! \left(x \right)+F_{19}\! \left(x \right)\\ F_{119}\! \left(x \right) &= F_{115}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{120}\! \left(x \right) &= F_{121}\! \left(x \right) F_{19}\! \left(x \right)\\ F_{121}\! \left(x \right) &= F_{122}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{122}\! \left(x \right) &= F_{123}\! \left(x \right)+F_{13}\! \left(x \right)+F_{42}\! \left(x \right)\\ F_{123}\! \left(x \right) &= F_{121}\! \left(x \right) F_{8}\! \left(x \right)\\ \end{align*}\)

This specification was found using the strategy pack "Insertion Point Placements Req Corrob Expand Verified" and has 193 rules.

Found on January 21, 2022.

Finding the specification took 55 seconds.

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\(\begin{align*} F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{1}\! \left(x \right) &= 1\\ F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{4}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{4}\! \left(x \right) &= x\\ F_{5}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{4}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{13}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{10}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{17}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{128}\! \left(x \right)+F_{22}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{39}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{25}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{27}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{26}\! \left(x \right) &= 0\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{33}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{34}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{35}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{37}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{38}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{36}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{45}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{42}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{11}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{11}\! \left(x \right) F_{43}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{10} \left(x \right)^{2} F_{11}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{64}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{4}\! \left(x \right) F_{48}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)+F_{58}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)+F_{51}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{46}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{11}\! \left(x \right) F_{52}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{53}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{54}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{55}\! \left(x \right)+F_{57}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{4}\! \left(x \right) F_{56}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{54}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{4}\! \left(x \right) F_{53}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{11}\! \left(x \right) F_{59}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{60}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{61}\! \left(x \right)+F_{63}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{4}\! \left(x \right) F_{62}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{60}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{4}\! \left(x \right) F_{59}\! \left(x \right)\\ F_{64}\! \left(x \right) &= -F_{112}\! \left(x \right)+F_{65}\! \left(x \right)\\ F_{65}\! \left(x \right) &= -F_{107}\! \left(x \right)+F_{66}\! \left(x \right)\\ F_{66}\! \left(x \right) &= -F_{106}\! \left(x \right)+F_{67}\! \left(x \right)\\ F_{67}\! \left(x \right) &= \frac{F_{68}\! \left(x \right)}{F_{4}\! \left(x \right)}\\ F_{68}\! \left(x \right) &= F_{69}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{70}\! \left(x \right)+F_{96}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{4}\! \left(x \right) F_{71}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{72}\! \left(x \right)+F_{79}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{7}\! \left(x \right)+F_{73}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{74}\! \left(x \right)+F_{75}\! \left(x \right)\\ F_{74}\! \left(x \right) &= F_{11}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{75}\! \left(x \right) &= F_{4}\! \left(x \right) F_{76}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{77}\! \left(x \right)+F_{78}\! \left(x \right)\\ F_{77}\! \left(x \right) &= F_{4}\! \left(x \right)\\ F_{78}\! \left(x \right) &= F_{73}\! \left(x \right)\\ F_{79}\! \left(x \right) &= F_{69}\! \left(x \right)+F_{80}\! \left(x \right)\\ F_{80}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{81}\! \left(x \right)+F_{85}\! \left(x \right)+F_{89}\! \left(x \right)\\ F_{81}\! \left(x \right) &= F_{4}\! \left(x \right) F_{82}\! \left(x \right)\\ F_{82}\! \left(x \right) &= F_{83}\! \left(x \right)+F_{84}\! \left(x \right)\\ F_{83}\! \left(x \right) &= F_{73}\! \left(x \right)\\ F_{84}\! \left(x \right) &= F_{80}\! \left(x \right)\\ F_{85}\! \left(x \right) &= F_{4}\! \left(x \right) F_{86}\! \left(x \right)\\ F_{86}\! \left(x \right) &= F_{87}\! \left(x \right)\\ F_{87}\! \left(x \right) &= F_{4}\! \left(x \right) F_{88}\! \left(x \right)\\ F_{88}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{86}\! \left(x \right)\\ F_{89}\! \left(x \right) &= F_{4}\! \left(x \right) F_{90}\! \left(x \right)\\ F_{90}\! \left(x \right) &= F_{91}\! \left(x \right)+F_{94}\! \left(x \right)\\ F_{91}\! \left(x \right) &= F_{92}\! \left(x \right)\\ F_{92}\! \left(x \right) &= F_{93}\! \left(x \right)\\ F_{93}\! \left(x \right) &= F_{11}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{94}\! \left(x \right) &= F_{95}\! \left(x \right)\\ F_{95}\! \left(x \right) &= 2 F_{26}\! \left(x \right)+F_{85}\! \left(x \right)+F_{89}\! \left(x \right)\\ F_{96}\! \left(x \right) &= F_{4}\! \left(x \right) F_{97}\! \left(x \right)\\ F_{97}\! \left(x \right) &= F_{98}\! \left(x \right)+F_{99}\! \left(x \right)\\ F_{98}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{86}\! \left(x \right)\\ F_{99}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{103}\! \left(x \right)\\ F_{100}\! \left(x \right) &= F_{101}\! \left(x \right)+F_{26}\! \left(x \right)+F_{96}\! \left(x \right)\\ F_{101}\! \left(x \right) &= F_{102}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{102}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{103}\! \left(x \right) &= F_{104}\! \left(x \right)\\ F_{104}\! \left(x \right) &= F_{105}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{105}\! \left(x \right) &= F_{103}\! \left(x \right)+F_{14}\! \left(x \right)\\ F_{106}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{72}\! \left(x \right)\\ F_{107}\! \left(x \right) &= F_{108}\! \left(x \right)+F_{40}\! \left(x \right)\\ F_{108}\! \left(x \right) &= F_{109}\! \left(x \right)\\ F_{109}\! \left(x \right) &= F_{110} \left(x \right)^{2} F_{4}\! \left(x \right)\\ F_{110}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{111}\! \left(x \right)\\ F_{111}\! \left(x \right) &= F_{10}\! \left(x \right) F_{11}\! \left(x \right)\\ F_{112}\! \left(x \right) &= F_{113}\! \left(x \right)+F_{26}\! \left(x \right)+F_{57}\! \left(x \right)\\ F_{113}\! \left(x \right) &= F_{114}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{114}\! \left(x \right) &= F_{115}\! \left(x \right)+F_{119}\! \left(x \right)\\ F_{115}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{116}\! \left(x \right)\\ F_{116}\! \left(x \right) &= F_{117}\! \left(x \right)\\ F_{117}\! \left(x \right) &= F_{118}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{118}\! \left(x \right) &= F_{116}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{119}\! \left(x \right) &= F_{112}\! \left(x \right)+F_{120}\! \left(x \right)\\ F_{120}\! \left(x \right) &= 2 F_{26}\! \left(x \right)+F_{121}\! \left(x \right)+F_{125}\! \left(x \right)\\ F_{121}\! \left(x \right) &= F_{122}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{122}\! \left(x \right) &= F_{123}\! \left(x \right)+F_{124}\! \left(x \right)\\ F_{123}\! \left(x \right) &= F_{116}\! \left(x \right)\\ F_{124}\! \left(x \right) &= F_{120}\! \left(x \right)\\ F_{125}\! \left(x \right) &= F_{126}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{126}\! \left(x \right) &= F_{127}\! \left(x \right)+F_{92}\! \left(x \right)\\ F_{127}\! \left(x \right) &= F_{125}\! \left(x \right)\\ F_{128}\! \left(x \right) &= F_{129}\! \left(x \right)+F_{139}\! \left(x \right)\\ F_{129}\! \left(x \right) &= F_{130}\! \left(x \right)\\ F_{130}\! \left(x \right) &= F_{131}\! \left(x \right)+F_{138}\! \left(x \right)\\ F_{131}\! \left(x \right) &= F_{11}\! \left(x \right) F_{132}\! \left(x \right)\\ F_{132}\! \left(x \right) &= F_{133}\! \left(x \right)\\ F_{133}\! \left(x \right) &= F_{134}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{134}\! \left(x \right) &= F_{135}\! \left(x \right)+F_{136}\! \left(x \right)\\ F_{135}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{132}\! \left(x \right)\\ F_{136}\! \left(x \right) &= F_{137}\! \left(x \right)\\ F_{137}\! \left(x \right) &= F_{10} \left(x \right)^{2} F_{4}\! \left(x \right)\\ F_{138}\! \left(x \right) &= F_{11} \left(x \right)^{2} F_{10}\! \left(x \right)\\ F_{139}\! \left(x \right) &= F_{140}\! \left(x \right)+F_{142}\! \left(x \right)\\ F_{140}\! \left(x \right) &= F_{141}\! \left(x \right)\\ F_{141}\! \left(x \right) &= F_{11} \left(x \right)^{3}\\ F_{142}\! \left(x \right) &= F_{143}\! \left(x \right)+F_{191}\! \left(x \right)\\ F_{143}\! \left(x \right) &= F_{144}\! \left(x \right)\\ F_{144}\! \left(x \right) &= F_{145}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{145}\! \left(x \right) &= F_{146}\! \left(x \right)+F_{158}\! \left(x \right)\\ F_{146}\! \left(x \right) &= F_{143}\! \left(x \right)+F_{147}\! \left(x \right)\\ F_{147}\! \left(x \right) &= F_{148}\! \left(x \right)\\ F_{148}\! \left(x \right) &= F_{149}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{149}\! \left(x \right) &= F_{150}\! \left(x \right)+F_{152}\! \left(x \right)\\ F_{150}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{151}\! \left(x \right)\\ F_{151}\! \left(x \right) &= F_{74}\! \left(x \right)\\ F_{152}\! \left(x \right) &= F_{147}\! \left(x \right)+F_{153}\! \left(x \right)\\ F_{153}\! \left(x \right) &= 2 F_{26}\! \left(x \right)+F_{154}\! \left(x \right)+F_{85}\! \left(x \right)\\ F_{154}\! \left(x \right) &= F_{155}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{155}\! \left(x \right) &= F_{156}\! \left(x \right)+F_{157}\! \left(x \right)\\ F_{156}\! \left(x \right) &= F_{151}\! \left(x \right)\\ F_{157}\! \left(x \right) &= F_{153}\! \left(x \right)\\ F_{158}\! \left(x \right) &= F_{159}\! \left(x \right)+F_{166}\! \left(x \right)\\ F_{159}\! \left(x \right) &= F_{160}\! \left(x \right)+F_{162}\! \left(x \right)\\ F_{160}\! \left(x \right) &= F_{161}\! \left(x \right)\\ F_{161}\! \left(x \right) &= F_{10} \left(x \right)^{2} F_{11} \left(x \right)^{2}\\ F_{162}\! \left(x \right) &= F_{163}\! \left(x \right)\\ F_{163}\! \left(x \right) &= F_{10}\! \left(x \right) F_{11}\! \left(x \right) F_{164}\! \left(x \right)\\ F_{164}\! \left(x \right) &= F_{165}\! \left(x \right)+F_{43}\! \left(x \right)\\ F_{165}\! \left(x \right) &= F_{10}\! \left(x \right) F_{11}\! \left(x \right)\\ F_{166}\! \left(x \right) &= F_{11}\! \left(x \right) F_{167}\! \left(x \right)\\ F_{167}\! \left(x \right) &= -F_{190}\! \left(x \right)+F_{168}\! \left(x \right)\\ F_{168}\! \left(x \right) &= -F_{59}\! \left(x \right)+F_{169}\! \left(x \right)\\ F_{169}\! \left(x \right) &= \frac{F_{170}\! \left(x \right)}{F_{4}\! \left(x \right)}\\ F_{170}\! \left(x \right) &= F_{171}\! \left(x \right)\\ F_{171}\! \left(x \right) &= F_{172}\! \left(x \right)+F_{26}\! \left(x \right)+F_{63}\! \left(x \right)\\ F_{172}\! \left(x \right) &= F_{173}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{173}\! \left(x \right) &= F_{174}\! \left(x \right)+F_{178}\! \left(x \right)\\ F_{174}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{175}\! \left(x \right)\\ F_{175}\! \left(x \right) &= F_{176}\! \left(x \right)+F_{26}\! \left(x \right)+F_{74}\! \left(x \right)\\ F_{176}\! \left(x \right) &= F_{177}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{177}\! \left(x \right) &= F_{175}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{178}\! \left(x \right) &= F_{171}\! \left(x \right)+F_{179}\! \left(x \right)\\ F_{179}\! \left(x \right) &= F_{180}\! \left(x \right)+F_{184}\! \left(x \right)+F_{185}\! \left(x \right)+F_{26}\! \left(x \right)\\ F_{180}\! \left(x \right) &= F_{181}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{181}\! \left(x \right) &= F_{182}\! \left(x \right)+F_{183}\! \left(x \right)\\ F_{182}\! \left(x \right) &= F_{175}\! \left(x \right)\\ F_{183}\! \left(x \right) &= F_{179}\! \left(x \right)\\ F_{184}\! \left(x \right) &= F_{4}\! \left(x \right) F_{60}\! \left(x \right)\\ F_{185}\! \left(x \right) &= F_{186}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{186}\! \left(x \right) &= F_{187}\! \left(x \right)+F_{189}\! \left(x \right)\\ F_{187}\! \left(x \right) &= F_{188}\! \left(x \right)\\ F_{188}\! \left(x \right) &= F_{11}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{189}\! \left(x \right) &= 2 F_{26}\! \left(x \right)+F_{184}\! \left(x \right)+F_{185}\! \left(x \right)\\ F_{190}\! \left(x \right) &= F_{10}\! \left(x \right) F_{132}\! \left(x \right)\\ F_{191}\! \left(x \right) &= F_{192}\! \left(x \right)\\ F_{192}\! \left(x \right) &= F_{11} \left(x \right)^{4}\\ \end{align*}\)

This specification was found using the strategy pack "Insertion Point Row And Col Placements Req Corrob Expand Verified" and has 156 rules.

Found on January 21, 2022.

Finding the specification took 43 seconds.

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Copy 156 equations to clipboard:
\(\begin{align*} F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{1}\! \left(x \right) &= 1\\ F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{13}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{13}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{5}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{13}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{14}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{10}\! \left(x \right) F_{13}\! \left(x \right)\\ F_{13}\! \left(x \right) &= x\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{10}\! \left(x \right) F_{13}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{24}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{18}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{19}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{21}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{20}\! \left(x \right) &= 0\\ F_{21}\! \left(x \right) &= F_{13}\! \left(x \right) F_{22}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{19}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{13}\! \left(x \right) F_{18}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{25}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{26}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{13}\! \left(x \right) F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{25}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{13}\! \left(x \right) F_{24}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{41}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{13}\! \left(x \right) F_{32}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{34}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{13}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{35}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{36}\! \left(x \right)+F_{40}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{13}\! \left(x \right) F_{37}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{39}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{13}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{35}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{11}\! \left(x \right) F_{13}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{102}\! \left(x \right)+F_{20}\! \left(x \right)+F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{13}\! \left(x \right) F_{43}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{59}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{53}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{13}\! \left(x \right) F_{47}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{48}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{49}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{13}\! \left(x \right) F_{51}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{52}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{50}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{54}\! \left(x \right)+F_{55}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{11}\! \left(x \right) F_{13}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{13}\! \left(x \right) F_{56}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)+F_{58}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{13}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{53}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{60}\! \left(x \right)+F_{88}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{61}\! \left(x \right)+F_{78}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{13}\! \left(x \right) F_{62}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{63}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{60}\! \left(x \right)+F_{64}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{65}\! \left(x \right)+F_{69}\! \left(x \right)+F_{73}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{13}\! \left(x \right) F_{66}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{67}\! \left(x \right)+F_{68}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{53}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{64}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{13}\! \left(x \right) F_{70}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{71}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{13}\! \left(x \right) F_{72}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{70}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{13}\! \left(x \right) F_{74}\! \left(x \right)\\ F_{74}\! \left(x \right) &= F_{75}\! \left(x \right)+F_{76}\! \left(x \right)\\ F_{75}\! \left(x \right) &= F_{40}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{77}\! \left(x \right)\\ F_{77}\! \left(x \right) &= 2 F_{20}\! \left(x \right)+F_{69}\! \left(x \right)+F_{73}\! \left(x \right)\\ F_{78}\! \left(x \right) &= F_{13}\! \left(x \right) F_{79}\! \left(x \right)\\ F_{79}\! \left(x \right) &= F_{80}\! \left(x \right)+F_{81}\! \left(x \right)\\ F_{80}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{70}\! \left(x \right)\\ F_{81}\! \left(x \right) &= F_{82}\! \left(x \right)+F_{85}\! \left(x \right)\\ F_{82}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{78}\! \left(x \right)+F_{83}\! \left(x \right)\\ F_{83}\! \left(x \right) &= F_{13}\! \left(x \right) F_{84}\! \left(x \right)\\ F_{84}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{82}\! \left(x \right)\\ F_{85}\! \left(x \right) &= F_{86}\! \left(x \right)\\ F_{86}\! \left(x \right) &= F_{13}\! \left(x \right) F_{87}\! \left(x \right)\\ F_{87}\! \left(x \right) &= F_{49}\! \left(x \right)+F_{85}\! \left(x \right)\\ F_{88}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{69}\! \left(x \right)+F_{89}\! \left(x \right)+F_{92}\! \left(x \right)\\ F_{89}\! \left(x \right) &= F_{13}\! \left(x \right) F_{90}\! \left(x \right)\\ F_{90}\! \left(x \right) &= F_{67}\! \left(x \right)+F_{91}\! \left(x \right)\\ F_{91}\! \left(x \right) &= F_{88}\! \left(x \right)\\ F_{92}\! \left(x \right) &= F_{13}\! \left(x \right) F_{93}\! \left(x \right)\\ F_{93}\! \left(x \right) &= F_{94}\! \left(x \right)+F_{98}\! \left(x \right)\\ F_{94}\! \left(x \right) &= F_{95}\! \left(x \right)\\ F_{95}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{40}\! \left(x \right)+F_{96}\! \left(x \right)\\ F_{96}\! \left(x \right) &= F_{13}\! \left(x \right) F_{97}\! \left(x \right)\\ F_{97}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{95}\! \left(x \right)\\ F_{98}\! \left(x \right) &= F_{99}\! \left(x \right)\\ F_{99}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{20}\! \left(x \right)+F_{69}\! \left(x \right)+F_{92}\! \left(x \right)\\ F_{100}\! \left(x \right) &= F_{101}\! \left(x \right) F_{13}\! \left(x \right)\\ F_{101}\! \left(x \right) &= F_{53}\! \left(x \right)+F_{99}\! \left(x \right)\\ F_{102}\! \left(x \right) &= F_{103}\! \left(x \right) F_{13}\! \left(x \right)\\ F_{103}\! \left(x \right) &= F_{104}\! \left(x \right)+F_{116}\! \left(x \right)\\ F_{104}\! \left(x \right) &= F_{105}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{105}\! \left(x \right) &= F_{106}\! \left(x \right)\\ F_{106}\! \left(x \right) &= F_{107}\! \left(x \right) F_{13}\! \left(x \right)\\ F_{107}\! \left(x \right) &= F_{108}\! \left(x \right)+F_{110}\! \left(x \right)\\ F_{108}\! \left(x \right) &= F_{109}\! \left(x \right)+F_{11}\! \left(x \right)\\ F_{109}\! \left(x \right) &= F_{54}\! \left(x \right)\\ F_{110}\! \left(x \right) &= F_{105}\! \left(x \right)+F_{111}\! \left(x \right)\\ F_{111}\! \left(x \right) &= 2 F_{20}\! \left(x \right)+F_{112}\! \left(x \right)+F_{69}\! \left(x \right)\\ F_{112}\! \left(x \right) &= F_{113}\! \left(x \right) F_{13}\! \left(x \right)\\ F_{113}\! \left(x \right) &= F_{114}\! \left(x \right)+F_{115}\! \left(x \right)\\ F_{114}\! \left(x \right) &= F_{109}\! \left(x \right)\\ F_{115}\! \left(x \right) &= F_{111}\! \left(x \right)\\ F_{116}\! \left(x \right) &= F_{117}\! \left(x \right)+F_{41}\! \left(x \right)\\ F_{117}\! \left(x \right) &= F_{118}\! \left(x \right)\\ F_{118}\! \left(x \right) &= F_{119}\! \left(x \right) F_{13}\! \left(x \right)\\ F_{119}\! \left(x \right) &= F_{120}\! \left(x \right)+F_{121}\! \left(x \right)\\ F_{120}\! \left(x \right) &= F_{105}\! \left(x \right)+F_{117}\! \left(x \right)\\ F_{121}\! \left(x \right) &= F_{122}\! \left(x \right)+F_{131}\! \left(x \right)\\ F_{122}\! \left(x \right) &= F_{123}\! \left(x \right)+F_{125}\! \left(x \right)\\ F_{123}\! \left(x \right) &= F_{124}\! \left(x \right)\\ F_{124}\! \left(x \right) &= F_{10} \left(x \right)^{2} F_{11} \left(x \right)^{2}\\ F_{125}\! \left(x \right) &= F_{126}\! \left(x \right)\\ F_{126}\! \left(x \right) &= F_{10}\! \left(x \right) F_{11}\! \left(x \right) F_{127}\! \left(x \right)\\ F_{127}\! \left(x \right) &= F_{128}\! \left(x \right)+F_{129}\! \left(x \right)\\ F_{128}\! \left(x \right) &= F_{10}\! \left(x \right) F_{11}\! \left(x \right)\\ F_{129}\! \left(x \right) &= F_{130}\! \left(x \right)\\ F_{130}\! \left(x \right) &= F_{10} \left(x \right)^{2} F_{11}\! \left(x \right) F_{13}\! \left(x \right)\\ F_{131}\! \left(x \right) &= F_{11}\! \left(x \right) F_{132}\! \left(x \right)\\ F_{132}\! \left(x \right) &= -F_{155}\! \left(x \right)+F_{133}\! \left(x \right)\\ F_{133}\! \left(x \right) &= -F_{18}\! \left(x \right)+F_{134}\! \left(x \right)\\ F_{134}\! \left(x \right) &= \frac{F_{135}\! \left(x \right)}{F_{13}\! \left(x \right)}\\ F_{135}\! \left(x \right) &= F_{136}\! \left(x \right)\\ F_{136}\! \left(x \right) &= F_{137}\! \left(x \right)+F_{20}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{137}\! \left(x \right) &= F_{13}\! \left(x \right) F_{138}\! \left(x \right)\\ F_{138}\! \left(x \right) &= F_{139}\! \left(x \right)+F_{143}\! \left(x \right)\\ F_{139}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{140}\! \left(x \right)\\ F_{140}\! \left(x \right) &= F_{141}\! \left(x \right)+F_{20}\! \left(x \right)+F_{54}\! \left(x \right)\\ F_{141}\! \left(x \right) &= F_{13}\! \left(x \right) F_{142}\! \left(x \right)\\ F_{142}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{140}\! \left(x \right)\\ F_{143}\! \left(x \right) &= F_{136}\! \left(x \right)+F_{144}\! \left(x \right)\\ F_{144}\! \left(x \right) &= F_{145}\! \left(x \right)+F_{149}\! \left(x \right)+F_{150}\! \left(x \right)+F_{20}\! \left(x \right)\\ F_{145}\! \left(x \right) &= F_{13}\! \left(x \right) F_{146}\! \left(x \right)\\ F_{146}\! \left(x \right) &= F_{147}\! \left(x \right)+F_{148}\! \left(x \right)\\ F_{147}\! \left(x \right) &= F_{140}\! \left(x \right)\\ F_{148}\! \left(x \right) &= F_{144}\! \left(x \right)\\ F_{149}\! \left(x \right) &= F_{13}\! \left(x \right) F_{19}\! \left(x \right)\\ F_{150}\! \left(x \right) &= F_{13}\! \left(x \right) F_{151}\! \left(x \right)\\ F_{151}\! \left(x \right) &= F_{152}\! \left(x \right)+F_{154}\! \left(x \right)\\ F_{152}\! \left(x \right) &= F_{153}\! \left(x \right)\\ F_{153}\! \left(x \right) &= F_{11}\! \left(x \right) F_{13}\! \left(x \right)\\ F_{154}\! \left(x \right) &= 2 F_{20}\! \left(x \right)+F_{149}\! \left(x \right)+F_{150}\! \left(x \right)\\ F_{155}\! \left(x \right) &= F_{10}\! \left(x \right) F_{30}\! \left(x \right)\\ \end{align*}\)

This specification was found using the strategy pack "Insertion Partial Point Placements Expand Verified" and has 282 rules.

Found on January 21, 2022.

Finding the specification took 232 seconds.

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Copy 282 equations to clipboard:
\(\begin{align*} F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{1}\! \left(x \right) &= 1\\ F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{4}\! \left(x \right) &= x\\ F_{5}\! \left(x \right) &= F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{4}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{14}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{12}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{11}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{18}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{16}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{41}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{24}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{21}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{24}\! \left(x \right)+F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{11}\! \left(x \right) F_{28}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{36}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{31}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{33}\! \left(x \right)+F_{35}\! \left(x \right)\\ F_{32}\! \left(x \right) &= 0\\ F_{33}\! \left(x \right) &= F_{34}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{31}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{30}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{37}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{38}\! \left(x \right)+F_{40}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{39}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{37}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{36}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{64}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{44}\! \left(x \right)+F_{45}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{4}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{4}\! \left(x \right) F_{46}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{51}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{48}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{49}\! \left(x \right)+F_{50}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{12}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{4}\! \left(x \right) F_{47}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)+F_{53}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{45}\! \left(x \right)+F_{49}\! \left(x \right)\\ F_{53}\! \left(x \right) &= 2 F_{32}\! \left(x \right)+F_{54}\! \left(x \right)+F_{58}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{4}\! \left(x \right) F_{55}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{4}\! \left(x \right) F_{57}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{55}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{4}\! \left(x \right) F_{59}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{60}\! \left(x \right)+F_{63}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{61}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{4}\! \left(x \right) F_{62}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{60}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{58}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{65}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{4}\! \left(x \right) F_{66}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{67}\! \left(x \right)+F_{77}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{68}\! \left(x \right)+F_{70}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{4}\! \left(x \right) F_{69}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{55}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{4}\! \left(x \right) F_{71}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{57}\! \left(x \right)+F_{72}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{67}\! \left(x \right)+F_{73}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{74}\! \left(x \right)\\ F_{74}\! \left(x \right) &= F_{4}\! \left(x \right) F_{75}\! \left(x \right)\\ F_{75}\! \left(x \right) &= F_{55}\! \left(x \right)+F_{76}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{74}\! \left(x \right)\\ F_{77}\! \left(x \right) &= F_{189}\! \left(x \right)+F_{78}\! \left(x \right)\\ F_{78}\! \left(x \right) &= F_{79}\! \left(x \right)+F_{84}\! \left(x \right)\\ F_{79}\! \left(x \right) &= F_{80}\! \left(x \right)+F_{81}\! \left(x \right)\\ F_{80}\! \left(x \right) &= F_{12}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{81}\! \left(x \right) &= F_{12}\! \left(x \right) F_{82}\! \left(x \right)\\ F_{82}\! \left(x \right) &= F_{83}\! \left(x \right)\\ F_{83}\! \left(x \right) &= F_{11} \left(x \right)^{2} F_{12}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{84}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{85}\! \left(x \right)\\ F_{85}\! \left(x \right) &= -F_{187}\! \left(x \right)+F_{86}\! \left(x \right)\\ F_{86}\! \left(x \right) &= \frac{F_{87}\! \left(x \right)}{F_{4}\! \left(x \right)}\\ F_{87}\! \left(x \right) &= F_{88}\! \left(x \right)\\ F_{88}\! \left(x \right) &= -F_{169}\! \left(x \right)+F_{89}\! \left(x \right)\\ F_{89}\! \left(x \right) &= -F_{134}\! \left(x \right)+F_{90}\! \left(x \right)\\ F_{90}\! \left(x \right) &= F_{91}\! \left(x \right)\\ F_{91}\! \left(x \right) &= F_{4}\! \left(x \right) F_{92}\! \left(x \right)\\ F_{92}\! \left(x \right) &= F_{128}\! \left(x \right)+F_{93}\! \left(x \right)\\ F_{93}\! \left(x \right) &= F_{116}\! \left(x \right)+F_{32}\! \left(x \right)+F_{94}\! \left(x \right)\\ F_{94}\! \left(x \right) &= F_{4}\! \left(x \right) F_{95}\! \left(x \right)\\ F_{95}\! \left(x \right) &= F_{102}\! \left(x \right)+F_{96}\! \left(x \right)\\ F_{96}\! \left(x \right) &= F_{8}\! \left(x \right)+F_{97}\! \left(x \right)\\ F_{97}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{49}\! \left(x \right)+F_{98}\! \left(x \right)\\ F_{98}\! \left(x \right) &= F_{4}\! \left(x \right) F_{99}\! \left(x \right)\\ F_{99}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{101}\! \left(x \right)\\ F_{100}\! \left(x \right) &= F_{4}\! \left(x \right)\\ F_{101}\! \left(x \right) &= F_{97}\! \left(x \right)\\ F_{102}\! \left(x \right) &= F_{103}\! \left(x \right)+F_{93}\! \left(x \right)\\ F_{103}\! \left(x \right) &= F_{104}\! \left(x \right)+F_{108}\! \left(x \right)+F_{109}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{104}\! \left(x \right) &= F_{105}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{105}\! \left(x \right) &= F_{106}\! \left(x \right)+F_{107}\! \left(x \right)\\ F_{106}\! \left(x \right) &= F_{97}\! \left(x \right)\\ F_{107}\! \left(x \right) &= F_{103}\! \left(x \right)\\ F_{108}\! \left(x \right) &= F_{31}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{109}\! \left(x \right) &= F_{110}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{110}\! \left(x \right) &= F_{111}\! \left(x \right)+F_{114}\! \left(x \right)\\ F_{111}\! \left(x \right) &= F_{112}\! \left(x \right)\\ F_{112}\! \left(x \right) &= F_{113}\! \left(x \right)\\ F_{113}\! \left(x \right) &= F_{12}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{114}\! \left(x \right) &= F_{115}\! \left(x \right)\\ F_{115}\! \left(x \right) &= 2 F_{32}\! \left(x \right)+F_{108}\! \left(x \right)+F_{109}\! \left(x \right)\\ F_{116}\! \left(x \right) &= F_{117}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{117}\! \left(x \right) &= F_{118}\! \left(x \right)+F_{122}\! \left(x \right)\\ F_{118}\! \left(x \right) &= F_{119}\! \left(x \right)+F_{12}\! \left(x \right)\\ F_{119}\! \left(x \right) &= F_{120}\! \left(x \right)\\ F_{120}\! \left(x \right) &= F_{121}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{121}\! \left(x \right) &= F_{119}\! \left(x \right)+F_{12}\! \left(x \right)\\ F_{122}\! \left(x \right) &= F_{123}\! \left(x \right)+F_{124}\! \left(x \right)\\ F_{123}\! \left(x \right) &= F_{116}\! \left(x \right)+F_{32}\! \left(x \right)+F_{33}\! \left(x \right)\\ F_{124}\! \left(x \right) &= F_{125}\! \left(x \right)\\ F_{125}\! \left(x \right) &= F_{126}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{126}\! \left(x \right) &= F_{127}\! \left(x \right)+F_{55}\! \left(x \right)\\ F_{127}\! \left(x \right) &= F_{125}\! \left(x \right)\\ F_{128}\! \left(x \right) &= F_{103}\! \left(x \right)+F_{129}\! \left(x \right)\\ F_{129}\! \left(x \right) &= F_{113}\! \left(x \right)+F_{130}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{130}\! \left(x \right) &= F_{131}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{131}\! \left(x \right) &= F_{132}\! \left(x \right)+F_{133}\! \left(x \right)\\ F_{132}\! \left(x \right) &= F_{4}\! \left(x \right)\\ F_{133}\! \left(x \right) &= F_{129}\! \left(x \right)\\ F_{134}\! \left(x \right) &= F_{135}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{135}\! \left(x \right) &= F_{136}\! \left(x \right)+F_{147}\! \left(x \right)\\ F_{136}\! \left(x \right) &= F_{137}\! \left(x \right)+F_{146}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{137}\! \left(x \right) &= F_{138}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{138}\! \left(x \right) &= F_{139}\! \left(x \right)+F_{142}\! \left(x \right)\\ F_{139}\! \left(x \right) &= F_{140}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{140}\! \left(x \right) &= F_{141}\! \left(x \right)\\ F_{141}\! \left(x \right) &= F_{12}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{142}\! \left(x \right) &= F_{143}\! \left(x \right)+F_{144}\! \left(x \right)\\ F_{143}\! \left(x \right) &= F_{141}\! \left(x \right)\\ F_{144}\! \left(x \right) &= F_{145}\! \left(x \right)\\ F_{145}\! \left(x \right) &= F_{18}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{146}\! \left(x \right) &= F_{4}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{147}\! \left(x \right) &= F_{148}\! \left(x \right)+F_{149}\! \left(x \right)\\ F_{148}\! \left(x \right) &= F_{137}\! \left(x \right)+F_{141}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{149}\! \left(x \right) &= F_{150}\! \left(x \right)+F_{160}\! \left(x \right)+F_{161}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{150}\! \left(x \right) &= F_{151}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{151}\! \left(x \right) &= F_{152}\! \left(x \right)+F_{154}\! \left(x \right)\\ F_{152}\! \left(x \right) &= F_{140}\! \left(x \right)+F_{153}\! \left(x \right)\\ F_{153}\! \left(x \right) &= F_{145}\! \left(x \right)\\ F_{154}\! \left(x \right) &= F_{144}\! \left(x \right)+F_{155}\! \left(x \right)\\ F_{155}\! \left(x \right) &= F_{156}\! \left(x \right)\\ F_{156}\! \left(x \right) &= F_{157}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{157}\! \left(x \right) &= F_{158}\! \left(x \right)\\ F_{158}\! \left(x \right) &= F_{159}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{159}\! \left(x \right) &= F_{157}\! \left(x \right)+F_{18}\! \left(x \right)\\ F_{160}\! \left(x \right) &= 0\\ F_{161}\! \left(x \right) &= F_{162}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{162}\! \left(x \right) &= F_{163}\! \left(x \right)+F_{165}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{163}\! \left(x \right) &= F_{164}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{164}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{18}\! \left(x \right)\\ F_{165}\! \left(x \right) &= F_{166}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{166}\! \left(x \right) &= F_{167}\! \left(x \right)+F_{168}\! \left(x \right)\\ F_{167}\! \left(x \right) &= F_{12}\! \left(x \right)\\ F_{168}\! \left(x \right) &= F_{162}\! \left(x \right)\\ F_{169}\! \left(x \right) &= -F_{174}\! \left(x \right)+F_{170}\! \left(x \right)\\ F_{170}\! \left(x \right) &= F_{171}\! \left(x \right)\\ F_{171}\! \left(x \right) &= F_{172}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{172}\! \left(x \right) &= F_{173}\! \left(x \right)+F_{79}\! \left(x \right)\\ F_{173}\! \left(x \right) &= F_{12} \left(x \right)^{2} F_{11}\! \left(x \right)\\ F_{174}\! \left(x \right) &= F_{175}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{175}\! \left(x \right) &= F_{176}\! \left(x \right)+F_{181}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{176}\! \left(x \right) &= F_{177}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{177}\! \left(x \right) &= F_{178}\! \left(x \right)+F_{179}\! \left(x \right)\\ F_{178}\! \left(x \right) &= F_{4}\! \left(x \right)\\ F_{179}\! \left(x \right) &= F_{180}\! \left(x \right)\\ F_{180}\! \left(x \right) &= F_{141}\! \left(x \right)+F_{176}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{181}\! \left(x \right) &= F_{182}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{182}\! \left(x \right) &= F_{183}\! \left(x \right)\\ F_{183}\! \left(x \right) &= F_{184}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{184}\! \left(x \right) &= F_{185}\! \left(x \right)+F_{186}\! \left(x \right)\\ F_{185}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{186}\! \left(x \right) &= F_{180}\! \left(x \right)+F_{182}\! \left(x \right)\\ F_{187}\! \left(x \right) &= F_{188}\! \left(x \right)\\ F_{188}\! \left(x \right) &= F_{12} \left(x \right)^{3} F_{11}\! \left(x \right)\\ F_{189}\! \left(x \right) &= -F_{280}\! \left(x \right)+F_{190}\! \left(x \right)\\ F_{190}\! \left(x \right) &= \frac{F_{191}\! \left(x \right)}{F_{4}\! \left(x \right)}\\ F_{191}\! \left(x \right) &= F_{192}\! \left(x \right)\\ F_{192}\! \left(x \right) &= -F_{272}\! \left(x \right)+F_{193}\! \left(x \right)\\ F_{193}\! \left(x \right) &= -F_{246}\! \left(x \right)+F_{194}\! \left(x \right)\\ F_{194}\! \left(x \right) &= F_{195}\! \left(x \right)\\ F_{195}\! \left(x \right) &= F_{196}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{196}\! \left(x \right) &= F_{197}\! \left(x \right)+F_{231}\! \left(x \right)\\ F_{197}\! \left(x \right) &= F_{198}\! \left(x \right)+F_{221}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{198}\! \left(x \right) &= F_{199}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{199}\! \left(x \right) &= F_{200}\! \left(x \right)+F_{206}\! \left(x \right)\\ F_{200}\! \left(x \right) &= F_{201}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{201}\! \left(x \right) &= F_{202}\! \left(x \right)+F_{32}\! \left(x \right)+F_{49}\! \left(x \right)\\ F_{202}\! \left(x \right) &= F_{203}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{203}\! \left(x \right) &= F_{204}\! \left(x \right)+F_{205}\! \left(x \right)\\ F_{204}\! \left(x \right) &= F_{4}\! \left(x \right)\\ F_{205}\! \left(x \right) &= F_{201}\! \left(x \right)\\ F_{206}\! \left(x \right) &= F_{197}\! \left(x \right)+F_{207}\! \left(x \right)\\ F_{207}\! \left(x \right) &= F_{208}\! \left(x \right)+F_{212}\! \left(x \right)+F_{216}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{208}\! \left(x \right) &= F_{209}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{209}\! \left(x \right) &= F_{210}\! \left(x \right)+F_{211}\! \left(x \right)\\ F_{210}\! \left(x \right) &= F_{201}\! \left(x \right)\\ F_{211}\! \left(x \right) &= F_{207}\! \left(x \right)\\ F_{212}\! \left(x \right) &= F_{213}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{213}\! \left(x \right) &= F_{214}\! \left(x \right)\\ F_{214}\! \left(x \right) &= F_{215}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{215}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{213}\! \left(x \right)\\ F_{216}\! \left(x \right) &= F_{217}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{217}\! \left(x \right) &= F_{218}\! \left(x \right)+F_{219}\! \left(x \right)\\ F_{218}\! \left(x \right) &= F_{140}\! \left(x \right)\\ F_{219}\! \left(x \right) &= F_{220}\! \left(x \right)\\ F_{220}\! \left(x \right) &= 2 F_{32}\! \left(x \right)+F_{212}\! \left(x \right)+F_{216}\! \left(x \right)\\ F_{221}\! \left(x \right) &= F_{222}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{222}\! \left(x \right) &= F_{223}\! \left(x \right)+F_{224}\! \left(x \right)\\ F_{223}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{213}\! \left(x \right)\\ F_{224}\! \left(x \right) &= F_{225}\! \left(x \right)+F_{228}\! \left(x \right)\\ F_{225}\! \left(x \right) &= F_{221}\! \left(x \right)+F_{226}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{226}\! \left(x \right) &= F_{227}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{227}\! \left(x \right) &= F_{225}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{228}\! \left(x \right) &= F_{229}\! \left(x \right)\\ F_{229}\! \left(x \right) &= F_{230}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{230}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{228}\! \left(x \right)\\ F_{231}\! \left(x \right) &= F_{180}\! \left(x \right)+F_{232}\! \left(x \right)\\ F_{232}\! \left(x \right) &= F_{212}\! \left(x \right)+F_{233}\! \left(x \right)+F_{236}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{233}\! \left(x \right) &= F_{234}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{234}\! \left(x \right) &= F_{210}\! \left(x \right)+F_{235}\! \left(x \right)\\ F_{235}\! \left(x \right) &= F_{232}\! \left(x \right)\\ F_{236}\! \left(x \right) &= F_{237}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{237}\! \left(x \right) &= F_{238}\! \left(x \right)+F_{242}\! \left(x \right)\\ F_{238}\! \left(x \right) &= F_{239}\! \left(x \right)\\ F_{239}\! \left(x \right) &= F_{141}\! \left(x \right)+F_{240}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{240}\! \left(x \right) &= F_{241}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{241}\! \left(x \right) &= F_{239}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{242}\! \left(x \right) &= F_{243}\! \left(x \right)\\ F_{243}\! \left(x \right) &= F_{212}\! \left(x \right)+F_{236}\! \left(x \right)+F_{244}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{244}\! \left(x \right) &= F_{245}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{245}\! \left(x \right) &= F_{201}\! \left(x \right)+F_{243}\! \left(x \right)\\ F_{246}\! \left(x \right) &= F_{247}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{247}\! \left(x \right) &= F_{248}\! \left(x \right)+F_{257}\! \left(x \right)\\ F_{248}\! \left(x \right) &= F_{249}\! \left(x \right)+F_{256}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{249}\! \left(x \right) &= F_{250}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{250}\! \left(x \right) &= F_{251}\! \left(x \right)+F_{252}\! \left(x \right)\\ F_{251}\! \left(x \right) &= F_{112}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{252}\! \left(x \right) &= F_{253}\! \left(x \right)+F_{254}\! \left(x \right)\\ F_{253}\! \left(x \right) &= F_{113}\! \left(x \right)\\ F_{254}\! \left(x \right) &= F_{255}\! \left(x \right)\\ F_{255}\! \left(x \right) &= F_{4}\! \left(x \right) F_{55}\! \left(x \right)\\ F_{256}\! \left(x \right) &= F_{4}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{257}\! \left(x \right) &= F_{258}\! \left(x \right)+F_{260}\! \left(x \right)\\ F_{258}\! \left(x \right) &= F_{113}\! \left(x \right)+F_{259}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{259}\! \left(x \right) &= F_{251}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{260}\! \left(x \right) &= F_{261}\! \left(x \right)+F_{264}\! \left(x \right)+F_{271}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{261}\! \left(x \right) &= F_{262}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{262}\! \left(x \right) &= F_{112}\! \left(x \right)+F_{263}\! \left(x \right)\\ F_{263}\! \left(x \right) &= F_{255}\! \left(x \right)\\ F_{264}\! \left(x \right) &= F_{265}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{265}\! \left(x \right) &= F_{266}\! \left(x \right)+F_{267}\! \left(x \right)\\ F_{266}\! \left(x \right) &= F_{112}\! \left(x \right)+F_{263}\! \left(x \right)\\ F_{267}\! \left(x \right) &= F_{268}\! \left(x \right)+F_{269}\! \left(x \right)\\ F_{268}\! \left(x \right) &= F_{255}\! \left(x \right)\\ F_{269}\! \left(x \right) &= F_{270}\! \left(x \right)\\ F_{270}\! \left(x \right) &= F_{4}\! \left(x \right) F_{76}\! \left(x \right)\\ F_{271}\! \left(x \right) &= F_{4}\! \left(x \right) F_{67}\! \left(x \right)\\ F_{272}\! \left(x \right) &= -F_{277}\! \left(x \right)+F_{273}\! \left(x \right)\\ F_{273}\! \left(x \right) &= F_{274}\! \left(x \right)\\ F_{274}\! \left(x \right) &= F_{275}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{275}\! \left(x \right) &= F_{276}\! \left(x \right)+F_{79}\! \left(x \right)\\ F_{276}\! \left(x \right) &= F_{12} \left(x \right)^{2} F_{11}\! \left(x \right)\\ F_{277}\! \left(x \right) &= F_{278}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{278}\! \left(x \right) &= F_{130}\! \left(x \right)+F_{279}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{279}\! \left(x \right) &= F_{182}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{280}\! \left(x \right) &= F_{281}\! \left(x \right)\\ F_{281}\! \left(x \right) &= F_{12} \left(x \right)^{3} F_{11}\! \left(x \right)\\ \end{align*}\)