Av(1234, 1342, 3412)
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Generating Function
\(\displaystyle -\frac{17 x^{9}-95 x^{8}+269 x^{7}-463 x^{6}+521 x^{5}-393 x^{4}+198 x^{3}-64 x^{2}+12 x -1}{\left(2 x -1\right)^{3} \left(-1+x \right)^{7}}\)
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Counting Sequence
1, 1, 2, 6, 21, 74, 248, 780, 2309, 6483, 17407, 45028, 112921, 275964, 660030, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(2 x -1\right)^{3} \left(-1+x \right)^{7} F \! \left(x \right)+17 x^{9}-95 x^{8}+269 x^{7}-463 x^{6}+521 x^{5}-393 x^{4}+198 x^{3}-64 x^{2}+12 x -1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 74\)
\(\displaystyle a \! \left(6\right) = 248\)
\(\displaystyle a \! \left(7\right) = 780\)
\(\displaystyle a \! \left(8\right) = 2309\)
\(\displaystyle a \! \left(9\right) = 6483\)
\(\displaystyle a \! \left(n +3\right) = -\frac{n^{6}}{720}+\frac{n^{5}}{48}-\frac{29 n^{4}}{144}+\frac{47 n^{3}}{48}-\frac{647 n^{2}}{360}+8 a \! \left(n \right)-12 a \! \left(n +1\right)+6 a \! \left(n +2\right)+4 n -2, \quad n \geq 10\)
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Explicit Closed Form
\(\displaystyle \frac{\left(90 n^{2}+2790 n -28800\right) 2^{n}}{720}+\frac{n^{6}}{720}+\frac{n^{5}}{240}+\frac{29 n^{4}}{144}+\frac{17 n^{3}}{16}+\frac{2627 n^{2}}{360}+\frac{703 n}{30}+41\)
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This specification was found using the strategy pack "Point And Row And Col Placements Req Corrob Expand Verified" and has 56 rules.

Found on January 21, 2022.

Finding the specification took 46 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_{15}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{15}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right) F_{15}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{12}\! \left(x \right)+F_{16}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{13}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{12}\! \left(x \right) F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= x\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{12}\! \left(x \right) F_{15}\! \left(x \right) F_{18}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{26}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{20}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{23}\! \left(x \right)+F_{25}\! \left(x \right)\\ F_{22}\! \left(x \right) &= 0\\ F_{23}\! \left(x \right) &= F_{15}\! \left(x \right) F_{24}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{21}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{15}\! \left(x \right) F_{20}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{28}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{15}\! \left(x \right) F_{29}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{15}\! \left(x \right) F_{26}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{32}\! \left(x \right)+F_{52}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{15}\! \left(x \right) F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{15}\! \left(x \right) F_{35}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{36}\! \left(x \right)+F_{42}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{15}\! \left(x \right) F_{37}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{41}\! \left(x \right)\\ F_{38}\! \left(x \right) &= \frac{F_{39}\! \left(x \right)}{F_{15}\! \left(x \right)}\\ F_{39}\! \left(x \right) &= -F_{40}\! \left(x \right)+F_{11}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{26}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{26}\! \left(x \right) F_{40}\! \left(x \right)\\ F_{42}\! \left(x \right) &= -F_{22}\! \left(x \right)-F_{47}\! \left(x \right)+F_{43}\! \left(x \right)\\ F_{43}\! \left(x \right) &= -F_{46}\! \left(x \right)+F_{44}\! \left(x \right)\\ F_{44}\! \left(x \right) &= \frac{F_{45}\! \left(x \right)}{F_{15}\! \left(x \right)}\\ F_{45}\! \left(x \right) &= F_{31}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{19}\! \left(x \right) F_{6}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{15}\! \left(x \right) F_{48}\! \left(x \right)\\ F_{48}\! \left(x \right) &= -F_{51}\! \left(x \right)+F_{49}\! \left(x \right)\\ F_{49}\! \left(x \right) &= \frac{F_{50}\! \left(x \right)}{F_{15}\! \left(x \right)}\\ F_{50}\! \left(x \right) &= F_{31}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{12}\! \left(x \right) F_{6}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{15}\! \left(x \right) F_{53}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{5}\! \left(x \right)+F_{54}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{12}\! \left(x \right) F_{15}\! \left(x \right) F_{19}\! \left(x \right) F_{6}\! \left(x \right)\\ \end{align*}\)

This specification was found using the strategy pack "Insertion Row And Col Placements Expand Verified" and has 121 rules.

Found on January 21, 2022.

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

This specification was found using the strategy pack "Row And Col Placements Expand Verified" and has 93 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_{8}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{31}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{5}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{6}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{6}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= x\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right) F_{12}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{15}\! \left(x \right)+F_{16}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{13}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{12}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{11}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{12}\! \left(x \right) F_{18}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{26}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{20}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{23}\! \left(x \right)+F_{25}\! \left(x \right)\\ F_{22}\! \left(x \right) &= 0\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{21}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{20}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{28}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{26}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{3}\! \left(x \right)+F_{33}\! \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_{36}\! \left(x \right)+F_{86}\! \left(x \right)\\ F_{36}\! \left(x \right) &= \frac{F_{37}\! \left(x \right)}{F_{8}\! \left(x \right)}\\ F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)\\ F_{38}\! \left(x \right) &= -F_{83}\! \left(x \right)+F_{39}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{42}\! \left(x \right)\\ F_{40}\! \left(x \right) &= \frac{F_{41}\! \left(x \right)}{F_{8}\! \left(x \right)}\\ F_{41}\! \left(x \right) &= -F_{1}\! \left(x \right)+F_{0}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{43}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{81}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{71}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{12}\! \left(x \right) F_{46}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{57}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{48}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{50}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{12}\! \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_{53}\! \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_{13}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{52}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{58}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{59}\! \left(x \right)+F_{61}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{60}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{48}\! \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_{20}\! \left(x \right)+F_{63}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{58}\! \left(x \right)+F_{64}\! \left(x \right)\\ F_{64}\! \left(x \right) &= 2 F_{22}\! \left(x \right)+F_{65}\! \left(x \right)+F_{67}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{66}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{52}\! \left(x \right)+F_{64}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{68}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{69}\! \left(x \right)+F_{70}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{21}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{64}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{19}\! \left(x \right) F_{72}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{73}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{74}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{74}\! \left(x \right) &= F_{72}\! \left(x \right)+F_{75}\! \left(x \right)\\ F_{75}\! \left(x \right) &= -F_{78}\! \left(x \right)+F_{76}\! \left(x \right)\\ F_{76}\! \left(x \right) &= \frac{F_{77}\! \left(x \right)}{F_{8}\! \left(x \right)}\\ F_{77}\! \left(x \right) &= -F_{1}\! \left(x \right)+F_{78}\! \left(x \right)\\ F_{78}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{79}\! \left(x \right)\\ F_{79}\! \left(x \right) &= F_{8}\! \left(x \right) F_{80}\! \left(x \right)\\ F_{80}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{78}\! \left(x \right)\\ F_{81}\! \left(x \right) &= F_{19}\! \left(x \right) F_{8}\! \left(x \right) F_{82}\! \left(x \right)\\ F_{82}\! \left(x \right) &= \frac{F_{75}\! \left(x \right)}{F_{8}\! \left(x \right)}\\ F_{83}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{84}\! \left(x \right)\\ F_{84}\! \left(x \right) &= F_{12}\! \left(x \right) F_{85}\! \left(x \right)\\ F_{85}\! \left(x \right) &= F_{77}\! \left(x \right)\\ F_{86}\! \left(x \right) &= F_{87}\! \left(x \right)\\ F_{87}\! \left(x \right) &= F_{19}\! \left(x \right) F_{8}\! \left(x \right) F_{88}\! \left(x \right)\\ F_{88}\! \left(x \right) &= \frac{F_{89}\! \left(x \right)}{F_{8}\! \left(x \right)}\\ F_{89}\! \left(x \right) &= F_{90}\! \left(x \right)\\ F_{90}\! \left(x \right) &= -F_{12}\! \left(x \right)-F_{91}\! \left(x \right)+F_{11}\! \left(x \right)\\ F_{91}\! \left(x \right) &= F_{92}\! \left(x \right)\\ F_{92}\! \left(x \right) &= F_{12}\! \left(x \right) F_{19}\! \left(x \right) F_{8}\! \left(x \right)\\ \end{align*}\)

This specification was found using the strategy pack "Point And Row Placements Expand Verified" and has 80 rules.

Found on January 21, 2022.

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

This specification was found using the strategy pack "Insertion Point Placements Req Corrob Expand Verified" and has 81 rules.

Found on January 21, 2022.

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