Av(1234, 1342, 4123)
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Generating Function
\(\displaystyle \frac{\left(-6 x^{4}+19 x^{3}-18 x^{2}+7 x -1\right) \sqrt{1-4 x}-2 x^{5}+22 x^{4}-43 x^{3}+30 x^{2}-9 x +1}{4 x^{5}-6 x^{4}+2 x^{3}}\)
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Counting Sequence
1, 1, 2, 6, 21, 76, 275, 991, 3566, 12850, 46458, 168686, 615340, 2255101, 8301270, ...
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Implicit Equation for the Generating Function
\(\displaystyle x^{3} \left(2 x -1\right)^{2} \left(x -1\right)^{2} F \left(x \right)^{2}+\left(2 x -1\right) \left(x -1\right) \left(2 x^{5}-22 x^{4}+43 x^{3}-30 x^{2}+9 x -1\right) F \! \left(x \right)+x^{7}+14 x^{6}-73 x^{5}+131 x^{4}-111 x^{3}+49 x^{2}-11 x +1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 76\)
\(\displaystyle a \! \left(6\right) = 275\)
\(\displaystyle a \! \left(7\right) = 991\)
\(\displaystyle a \! \left(n +7\right) = \frac{24 \left(2 n +1\right) a \! \left(n \right)}{10+n}-\frac{\left(447 n +1982\right) a \! \left(3+n \right)}{10+n}-\frac{4 \left(59 n +110\right) a \! \left(n +1\right)}{10+n}+\frac{2 \left(226 n +709\right) a \! \left(n +2\right)}{10+n}+\frac{\left(1450+251 n \right) a \! \left(n +4\right)}{10+n}-\frac{\left(81 n +580\right) a \! \left(n +5\right)}{10+n}+\frac{2 \left(7 n +60\right) a \! \left(n +6\right)}{10+n}, \quad n \geq 8\)
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This specification was found using the strategy pack "Row And Col Placements Tracked Fusion Isolated Single Fusion" and has 49 rules.

Found on April 20, 2021.

Finding the specification took 375 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_{4}\! \left(x \right)\\ F_{3}\! \left(x \right) &= x\\ F_{4}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{48}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{3}\! \left(x \right) F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{7}\! \left(x , 1\right)\\ F_{7}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x , y\right)+F_{12}\! \left(x , y\right)+F_{8}\! \left(x , y\right)\\ F_{8}\! \left(x , y\right) &= F_{7}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\ F_{9}\! \left(x , y\right) &= y x\\ F_{10}\! \left(x , y\right) &= F_{11}\! \left(x , y\right) F_{3}\! \left(x \right)\\ F_{11}\! \left(x , y\right) &= \frac{y F_{7}\! \left(x , y\right)-F_{7}\! \left(x , 1\right)}{-1+y}\\ F_{12}\! \left(x , y\right) &= F_{13}\! \left(x , y\right) F_{3}\! \left(x \right)\\ F_{13}\! \left(x , y\right) &= F_{14}\! \left(x \right)+F_{46}\! \left(x , y\right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{20}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x , 1\right)\\ F_{16}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{17}\! \left(x , y\right)+F_{19}\! \left(x , y\right)\\ F_{17}\! \left(x , y\right) &= F_{18}\! \left(x , y\right) F_{3}\! \left(x \right)\\ F_{18}\! \left(x , y\right) &= \frac{y F_{16}\! \left(x , y\right)-F_{16}\! \left(x , 1\right)}{-1+y}\\ F_{19}\! \left(x , y\right) &= F_{16}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right) F_{3}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{21}\! \left(x \right)+F_{23}\! \left(x \right)+F_{44}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{15}\! \left(x \right) F_{25}\! \left(x \right) F_{3}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{36}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{27}\! \left(x \right) F_{3}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{28}\! \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_{3}\! \left(x \right) F_{34}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{31}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{3}\! \left(x \right) F_{30}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{26}\! \left(x \right) F_{3}\! \left(x \right) F_{38}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{39}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{41}\! \left(x \right)+F_{43}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{3}\! \left(x \right) F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{40}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{3}\! \left(x \right) F_{39}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{25}\! \left(x \right) F_{27}\! \left(x \right) F_{3}\! \left(x \right)\\ F_{46}\! \left(x , y\right) &= F_{47}\! \left(x , y\right)\\ F_{47}\! \left(x , y\right) &= F_{13}\! \left(x , y\right) F_{27}\! \left(x \right) F_{9}\! \left(x , y\right)\\ F_{48}\! \left(x \right) &= F_{14}\! \left(x \right) F_{3}\! \left(x \right)\\ \end{align*}\)

This specification was found using the strategy pack "Col Placements Tracked Fusion Isolated" and has 113 rules.

Found on April 20, 2021.

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

This specification was found using the strategy pack "Point And Col Placements Tracked Fusion Isolated" and has 58 rules.

Found on April 20, 2021.

Finding the specification took 208 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_{4}\! \left(x \right)\\ F_{3}\! \left(x \right) &= x\\ F_{4}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{5}\! \left(x \right)+F_{57}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{3}\! \left(x \right) F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{7}\! \left(x , 1\right)\\ F_{7}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x , y\right)+F_{12}\! \left(x , y\right)+F_{8}\! \left(x , y\right)\\ F_{8}\! \left(x , y\right) &= F_{7}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\ F_{9}\! \left(x , y\right) &= y x\\ F_{10}\! \left(x , y\right) &= F_{11}\! \left(x , y\right) F_{3}\! \left(x \right)\\ F_{11}\! \left(x , y\right) &= -\frac{-y F_{7}\! \left(x , y\right)+F_{7}\! \left(x , 1\right)}{-1+y}\\ F_{12}\! \left(x , y\right) &= F_{13}\! \left(x , y\right) F_{3}\! \left(x \right)\\ F_{13}\! \left(x , y\right) &= F_{14}\! \left(x \right)+F_{46}\! \left(x , y\right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{20}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x , 1\right)\\ F_{16}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{17}\! \left(x , y\right)+F_{19}\! \left(x , y\right)\\ F_{17}\! \left(x , y\right) &= F_{18}\! \left(x , y\right) F_{3}\! \left(x \right)\\ F_{18}\! \left(x , y\right) &= -\frac{-y F_{16}\! \left(x , y\right)+F_{16}\! \left(x , 1\right)}{-1+y}\\ F_{19}\! \left(x , y\right) &= F_{16}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right) F_{3}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{21}\! \left(x \right)+F_{23}\! \left(x \right)+F_{44}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{15}\! \left(x \right) F_{25}\! \left(x \right) F_{3}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{36}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{27}\! \left(x \right) F_{3}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{28}\! \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_{3}\! \left(x \right) F_{34}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{31}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{3}\! \left(x \right) F_{30}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{26}\! \left(x \right) F_{3}\! \left(x \right) F_{38}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{39}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{41}\! \left(x \right)+F_{43}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{3}\! \left(x \right) F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{40}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{3}\! \left(x \right) F_{39}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{25}\! \left(x \right) F_{27}\! \left(x \right) F_{3}\! \left(x \right)\\ F_{46}\! \left(x , y\right) &= F_{47}\! \left(x , y\right)\\ F_{47}\! \left(x , y\right) &= F_{14}\! \left(x \right) F_{48}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\ F_{48}\! \left(x , y\right) &= F_{27}\! \left(x \right)+F_{49}\! \left(x , y\right)\\ F_{49}\! \left(x , y\right) &= F_{50}\! \left(x , y\right)+F_{53}\! \left(x , y\right)\\ F_{50}\! \left(x , y\right) &= F_{51}\! \left(x , y\right)\\ F_{51}\! \left(x , y\right) &= F_{52}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\ F_{52}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{50}\! \left(x , y\right)\\ F_{53}\! \left(x , y\right) &= F_{32}\! \left(x \right)+F_{54}\! \left(x , y\right)+F_{56}\! \left(x , y\right)\\ F_{54}\! \left(x , y\right) &= F_{55}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\ F_{55}\! \left(x , y\right) &= F_{28}\! \left(x \right)+F_{53}\! \left(x , y\right)\\ F_{56}\! \left(x , y\right) &= F_{3}\! \left(x \right) F_{49}\! \left(x , y\right)\\ F_{57}\! \left(x \right) &= F_{14}\! \left(x \right) F_{3}\! \left(x \right)\\ \end{align*}\)

This specification was found using the strategy pack "Col Placements Tracked Fusion Isolated Single Fusion" and has 119 rules.

Found on April 20, 2021.

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

This specification was found using the strategy pack "Col Placements Tracked Fusion" and has 109 rules.

Found on July 23, 2021.

Finding the specification took 11 seconds.

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