PermPal Database

Av(12354, 12453, 12543, 21354, 21453, 21543, 31254)

Counting Sequence

1, 1, 2, 6, 24, 113, 581, 3149, 17688, 102001, 600303, 3590921, 21768532, 133438243, 825696844, ...

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Implicit Equation for the Generating Function

\(\displaystyle \left(x -1\right) F \left(x \right)^{4}+\left(-x +2\right) F \left(x \right)^{3}+\left(-x -1\right) F \left(x \right)^{2}+F \! \left(x \right)-1 = 0\)

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Recurrence

\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 113\)
\(\displaystyle a(6) = 581\)
\(\displaystyle a(7) = 3149\)
\(\displaystyle a(8) = 17688\)
\(\displaystyle a(9) = 102001\)
\(\displaystyle a(10) = 600303\)
\(\displaystyle a(11) = 3590921\)
\(\displaystyle a{\left(n + 12 \right)} = \frac{1600 n \left(n + 1\right) \left(2 n + 1\right) a{\left(n \right)}}{8897 \left(n + 10\right) \left(n + 11\right) \left(n + 12\right)} + \frac{40 \left(n + 1\right) \left(1814 n^{2} + 4123 n + 2403\right) a{\left(n + 1 \right)}}{8897 \left(n + 10\right) \left(n + 11\right) \left(n + 12\right)} + \frac{\left(182641 n + 1732056\right) a{\left(n + 11 \right)}}{8897 \left(n + 12\right)} - \frac{2 \left(750095 n^{2} + 13435303 n + 60158832\right) a{\left(n + 10 \right)}}{8897 \left(n + 11\right) \left(n + 12\right)} - \frac{30 \left(2178 n^{3} - 12697 n^{2} - 81881 n - 96050\right) a{\left(n + 2 \right)}}{8897 \left(n + 10\right) \left(n + 11\right) \left(n + 12\right)} + \frac{2 \left(376384 n^{3} + 3722439 n^{2} + 12425549 n + 13912914\right) a{\left(n + 3 \right)}}{8897 \left(n + 10\right) \left(n + 11\right) \left(n + 12\right)} - \frac{12 \left(532131 n^{3} + 7201629 n^{2} + 32240951 n + 47807794\right) a{\left(n + 4 \right)}}{8897 \left(n + 10\right) \left(n + 11\right) \left(n + 12\right)} + \frac{2 \left(3278329 n^{3} + 83200071 n^{2} + 703342100 n + 1980771132\right) a{\left(n + 9 \right)}}{8897 \left(n + 10\right) \left(n + 11\right) \left(n + 12\right)} + \frac{4 \left(4749038 n^{3} + 75272808 n^{2} + 397041913 n + 697149048\right) a{\left(n + 5 \right)}}{8897 \left(n + 10\right) \left(n + 11\right) \left(n + 12\right)} - \frac{6 \left(4978800 n^{3} + 90328153 n^{2} + 546634963 n + 1103500758\right) a{\left(n + 6 \right)}}{8897 \left(n + 10\right) \left(n + 11\right) \left(n + 12\right)} - \frac{\left(17194963 n^{3} + 393821559 n^{2} + 3007073090 n + 7655825904\right) a{\left(n + 8 \right)}}{8897 \left(n + 10\right) \left(n + 11\right) \left(n + 12\right)} + \frac{\left(28463783 n^{3} + 583249173 n^{2} + 3987133012 n + 9093976596\right) a{\left(n + 7 \right)}}{8897 \left(n + 10\right) \left(n + 11\right) \left(n + 12\right)}, \quad n \geq 12\)

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This specification was found using the strategy pack "Point Placements Req Corrob" and has 161 rules.

Finding the specification took 31059 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_{14}\! \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_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{14}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{18}\! \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_{15}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right) F_{14}\! \left(x \right)\\ F_{12}\! \left(x \right) &= \frac{F_{13}\! \left(x \right)}{F_{14}\! \left(x \right)}\\ F_{13}\! \left(x \right) &= F_{5}\! \left(x \right)\\ F_{14}\! \left(x \right) &= x\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{14}\! \left(x \right) F_{17}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{13}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{31}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{14}\! \left(x \right) F_{22}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{14}\! \left(x \right) F_{24}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{26}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{22}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{12}\! \left(x \right) F_{14}\! \left(x \right) F_{28}\! \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_{14}\! \left(x \right) F_{28}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{14}\! \left(x \right) F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{160}\! \left(x \right)+F_{34}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{154}\! \left(x \right)+F_{35}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{48}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{2}\! \left(x \right) F_{37}\! \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_{39}\! \left(x \right) &= F_{14}\! \left(x \right) F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= -F_{46}\! \left(x \right)+F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= \frac{F_{42}\! \left(x \right)}{F_{14}\! \left(x \right)}\\ F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)\\ F_{43}\! \left(x \right) &= -F_{44}\! \left(x \right)+F_{12}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{45}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{10}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{12}\! \left(x \right) F_{14}\! \left(x \right) F_{28}\! \left(x \right)\\ F_{48}\! \left(x \right) &= -F_{152}\! \left(x \right)+F_{49}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{146}\! \left(x \right)+F_{50}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)+F_{53}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{14}\! \left(x \right) F_{46}\! \left(x \right)\\ F_{53}\! \left(x \right) &= -F_{144}\! \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_{56}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{134}\! \left(x \right)+F_{57}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{25}\! \left(x \right) F_{58}\! \left(x \right)\\ F_{58}\! \left(x \right) &= \frac{F_{59}\! \left(x \right)}{F_{126}\! \left(x \right)}\\ F_{59}\! \left(x \right) &= -F_{124}\! \left(x \right)+F_{60}\! \left(x \right)\\ F_{60}\! \left(x \right) &= \frac{F_{61}\! \left(x \right)}{F_{14}\! \left(x \right)}\\ F_{61}\! \left(x \right) &= F_{62}\! \left(x \right)\\ F_{62}\! \left(x \right) &= -F_{67}\! \left(x \right)+F_{63}\! \left(x \right)\\ F_{63}\! \left(x \right) &= \frac{F_{64}\! \left(x \right)}{F_{14}\! \left(x \right)}\\ F_{64}\! \left(x \right) &= F_{65}\! \left(x \right)\\ F_{65}\! \left(x \right) &= -F_{2}\! \left(x \right)+F_{66}\! \left(x \right)\\ F_{66}\! \left(x \right) &= -F_{0}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{105}\! \left(x \right)+F_{68}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{69}\! \left(x \right)+F_{70}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{0}\! \left(x \right) F_{2}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{71}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{14}\! \left(x \right) F_{72}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{73}\! \left(x \right)+F_{79}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{4}\! \left(x \right) F_{74}\! \left(x \right)\\ F_{74}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{75}\! \left(x \right)\\ F_{75}\! \left(x \right) &= -F_{2}\! \left(x \right)+F_{76}\! \left(x \right)\\ F_{76}\! \left(x \right) &= -F_{0}\! \left(x \right)+F_{77}\! \left(x \right)\\ F_{77}\! \left(x \right) &= \frac{F_{78}\! \left(x \right)}{F_{14}\! \left(x \right)}\\ F_{78}\! \left(x \right) &= F_{2}\! \left(x \right)\\ F_{79}\! \left(x \right) &= F_{80}\! \left(x \right)\\ F_{80}\! \left(x \right) &= F_{12}\! \left(x \right) F_{14}\! \left(x \right) F_{81}\! \left(x \right)\\ F_{81}\! \left(x \right) &= \frac{F_{82}\! \left(x \right)}{F_{14}\! \left(x \right) F_{88}\! \left(x \right)}\\ F_{82}\! \left(x \right) &= F_{83}\! \left(x \right)\\ F_{83}\! \left(x \right) &= -F_{86}\! \left(x \right)+F_{84}\! \left(x \right)\\ F_{84}\! \left(x \right) &= \frac{F_{85}\! \left(x \right)}{F_{14}\! \left(x \right)}\\ F_{85}\! \left(x \right) &= F_{75}\! \left(x \right)\\ F_{86}\! \left(x \right) &= F_{87}\! \left(x \right)+F_{89}\! \left(x \right)\\ F_{87}\! \left(x \right) &= F_{2}\! \left(x \right) F_{88}\! \left(x \right)\\ F_{88}\! \left(x \right) &= x^{2} F_{88} \left(x \right)^{2}-2 x F_{88} \left(x \right)^{2}+F_{88}\! \left(x \right) x +2 F_{88}\! \left(x \right)-1\\ F_{89}\! \left(x \right) &= F_{90}\! \left(x \right)\\ F_{90}\! \left(x \right) &= F_{14}\! \left(x \right) F_{91}\! \left(x \right)\\ F_{91}\! \left(x \right) &= F_{101}\! \left(x \right)+F_{92}\! \left(x \right)\\ F_{92}\! \left(x \right) &= F_{77}\! \left(x \right) F_{93}\! \left(x \right)\\ F_{93}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{94}\! \left(x \right)\\ F_{94}\! \left(x \right) &= -F_{100}\! \left(x \right)+F_{95}\! \left(x \right)\\ F_{95}\! \left(x \right) &= F_{96}\! \left(x \right)\\ F_{96}\! \left(x \right) &= F_{14}\! \left(x \right) F_{97}\! \left(x \right)\\ F_{97}\! \left(x \right) &= F_{83}\! \left(x \right)+F_{98}\! \left(x \right)\\ F_{98}\! \left(x \right) &= F_{86}\! \left(x \right)+F_{99}\! \left(x \right)\\ F_{99}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{95}\! \left(x \right)\\ F_{100}\! \left(x \right) &= x^{2} F_{100} \left(x \right)^{2}+2 x^{2} F_{100}\! \left(x \right)-2 x F_{100} \left(x \right)^{2}+x^{2}-3 x F_{100}\! \left(x \right)-x +2 F_{100}\! \left(x \right)\\ F_{101}\! \left(x \right) &= F_{102}\! \left(x \right)\\ F_{102}\! \left(x \right) &= F_{103}\! \left(x \right) F_{14}\! \left(x \right) F_{81}\! \left(x \right)\\ F_{103}\! \left(x \right) &= \frac{F_{104}\! \left(x \right)}{F_{14}\! \left(x \right)}\\ F_{104}\! \left(x \right) &= F_{93}\! \left(x \right)\\ F_{105}\! \left(x \right) &= F_{106}\! \left(x \right)\\ F_{106}\! \left(x \right) &= F_{107}\! \left(x \right) F_{14}\! \left(x \right)\\ F_{107}\! \left(x \right) &= F_{108}\! \left(x \right)+F_{112}\! \left(x \right)\\ F_{108}\! \left(x \right) &= F_{109}\! \left(x \right) F_{58}\! \left(x \right)\\ F_{109}\! \left(x \right) &= \frac{F_{110}\! \left(x \right)}{F_{14}\! \left(x \right)}\\ F_{110}\! \left(x \right) &= F_{111}\! \left(x \right)\\ F_{111}\! \left(x \right) &= -F_{100}\! \left(x \right)+F_{10}\! \left(x \right)\\ F_{112}\! \left(x \right) &= F_{113}\! \left(x \right)+F_{121}\! \left(x \right)\\ F_{113}\! \left(x \right) &= F_{114}\! \left(x \right)\\ F_{114}\! \left(x \right) &= F_{115}\! \left(x \right) F_{117}\! \left(x \right) F_{14}\! \left(x \right)\\ F_{115}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{116}\! \left(x \right)\\ F_{116}\! \left(x \right) &= F_{111}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{117}\! \left(x \right) &= F_{118}\! \left(x \right)+F_{119}\! \left(x \right)\\ F_{118}\! \left(x \right) &= F_{28}\! \left(x \right) F_{58}\! \left(x \right)\\ F_{119}\! \left(x \right) &= F_{120}\! \left(x \right)\\ F_{120}\! \left(x \right) &= F_{14}\! \left(x \right) F_{28}\! \left(x \right) F_{81}\! \left(x \right)\\ F_{121}\! \left(x \right) &= F_{122}\! \left(x \right)\\ F_{122}\! \left(x \right) &= F_{117}\! \left(x \right) F_{123}\! \left(x \right) F_{14}\! \left(x \right)\\ F_{123}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{43}\! \left(x \right)\\ F_{124}\! \left(x \right) &= F_{125}\! \left(x \right)\\ F_{125}\! \left(x \right) &= F_{117}\! \left(x \right) F_{14}\! \left(x \right)\\ F_{126}\! \left(x \right) &= \frac{F_{127}\! \left(x \right)}{F_{0}\! \left(x \right)}\\ F_{127}\! \left(x \right) &= -F_{130}\! \left(x \right)+F_{128}\! \left(x \right)\\ F_{128}\! \left(x \right) &= \frac{F_{129}\! \left(x \right)}{F_{14}\! \left(x \right)}\\ F_{129}\! \left(x \right) &= F_{95}\! \left(x \right)\\ F_{130}\! \left(x \right) &= -F_{133}\! \left(x \right)+F_{131}\! \left(x \right)\\ F_{131}\! \left(x \right) &= \frac{F_{132}\! \left(x \right)}{F_{14}\! \left(x \right)}\\ F_{132}\! \left(x \right) &= F_{94}\! \left(x \right)\\ F_{133}\! \left(x \right) &= F_{126}\! \left(x \right) F_{2}\! \left(x \right)\\ F_{134}\! \left(x \right) &= F_{135}\! \left(x \right)\\ F_{135}\! \left(x \right) &= -F_{143}\! \left(x \right)+F_{136}\! \left(x \right)\\ F_{136}\! \left(x \right) &= \frac{F_{137}\! \left(x \right)}{F_{14}\! \left(x \right)}\\ F_{137}\! \left(x \right) &= F_{138}\! \left(x \right)\\ F_{138}\! \left(x \right) &= F_{139}\! \left(x \right) F_{14}\! \left(x \right) F_{28}\! \left(x \right)\\ F_{139}\! \left(x \right) &= F_{140}\! \left(x \right)+F_{141}\! \left(x \right)\\ F_{140}\! \left(x \right) &= F_{12}\! \left(x \right) F_{28}\! \left(x \right) F_{58}\! \left(x \right)\\ F_{141}\! \left(x \right) &= F_{142}\! \left(x \right)\\ F_{142}\! \left(x \right) &= F_{117}\! \left(x \right) F_{12}\! \left(x \right) F_{14}\! \left(x \right)\\ F_{143}\! \left(x \right) &= F_{24}\! \left(x \right) F_{58}\! \left(x \right)\\ F_{144}\! \left(x \right) &= F_{145}\! \left(x \right)\\ F_{145}\! \left(x \right) &= F_{14}\! \left(x \right) F_{24}\! \left(x \right)\\ F_{146}\! \left(x \right) &= F_{147}\! \left(x \right)\\ F_{147}\! \left(x \right) &= -F_{51}\! \left(x \right)+F_{148}\! \left(x \right)\\ F_{148}\! \left(x \right) &= F_{149}\! \left(x \right)\\ F_{149}\! \left(x \right) &= F_{14}\! \left(x \right) F_{150}\! \left(x \right)\\ F_{150}\! \left(x \right) &= F_{151}\! \left(x \right)\\ F_{151}\! \left(x \right) &= F_{12}\! \left(x \right) F_{14}\! \left(x \right) F_{28}\! \left(x \right) F_{58}\! \left(x \right)\\ F_{152}\! \left(x \right) &= F_{153}\! \left(x \right)\\ F_{153}\! \left(x \right) &= F_{134}\! \left(x \right) F_{14}\! \left(x \right)\\ F_{154}\! \left(x \right) &= F_{155}\! \left(x \right)\\ F_{155}\! \left(x \right) &= F_{14}\! \left(x \right) F_{156}\! \left(x \right)\\ F_{156}\! \left(x \right) &= F_{157}\! \left(x \right)+F_{158}\! \left(x \right)\\ F_{157}\! \left(x \right) &= F_{117}\! \left(x \right) F_{40}\! \left(x \right)\\ F_{158}\! \left(x \right) &= \frac{F_{159}\! \left(x \right)}{F_{14}\! \left(x \right)}\\ F_{159}\! \left(x \right) &= F_{153}\! \left(x \right)\\ F_{160}\! \left(x \right) &= F_{51}\! \left(x \right)\\ \end{align*}\)

This specification was found using the strategy pack "Point Placements Req Corrob" and has 160 rules.

Finding the specification took 11007 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_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{15}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{45}\! \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_{42}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{12}\! \left(x \right)\\ F_{11}\! \left(x \right) &= x^{2} F_{11} \left(x \right)^{2}+2 x^{2} F_{11}\! \left(x \right)-2 x F_{11} \left(x \right)^{2}+x^{2}-3 x F_{11}\! \left(x \right)-x +2 F_{11}\! \left(x \right)\\ F_{12}\! \left(x \right) &= -F_{2}\! \left(x \right)+F_{13}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{15}\! \left(x \right) &= x\\ F_{16}\! \left(x \right) &= \frac{F_{17}\! \left(x \right)}{F_{15}\! \left(x \right)}\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\ F_{18}\! \left(x \right) &= -F_{26}\! \left(x \right)+F_{19}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{20}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{15}\! \left(x \right) F_{22}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{37}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{24}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{2}\! \left(x \right) F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= x^{2} F_{26} \left(x \right)^{2}-2 x F_{26} \left(x \right)^{2}+F_{26}\! \left(x \right) x +2 F_{26}\! \left(x \right)-1\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{15}\! \left(x \right) F_{29}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{33}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{18}\! \left(x \right) F_{31}\! \left(x \right)\\ F_{31}\! \left(x \right) &= \frac{F_{32}\! \left(x \right)}{F_{15}\! \left(x \right)}\\ F_{32}\! \left(x \right) &= F_{2}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{15}\! \left(x \right) F_{16}\! \left(x \right) F_{35}\! \left(x \right)\\ F_{35}\! \left(x \right) &= \frac{F_{36}\! \left(x \right)}{F_{15}\! \left(x \right) F_{26}\! \left(x \right)}\\ F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)\\ F_{37}\! \left(x \right) &= -F_{24}\! \left(x \right)+F_{38}\! \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_{40}\! \left(x \right) &= -F_{2}\! \left(x \right)+F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= -F_{0}\! \left(x \right)+F_{31}\! \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_{10}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{62}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{15}\! \left(x \right) F_{49}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{15}\! \left(x \right) F_{51}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)+F_{57}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)+F_{55}\! \left(x \right)\\ F_{53}\! \left(x \right) &= \frac{F_{54}\! \left(x \right)}{F_{15}\! \left(x \right)}\\ F_{54}\! \left(x \right) &= F_{5}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{15}\! \left(x \right) F_{51}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{15}\! \left(x \right) F_{53}\! \left(x \right) F_{59}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{60}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{61}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{15}\! \left(x \right) F_{59}\! \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_{159}\! \left(x \right)+F_{65}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{153}\! \left(x \right)+F_{66}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{67}\! \left(x \right)+F_{79}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{2}\! \left(x \right) F_{68}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{69}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{70}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{15}\! \left(x \right) F_{71}\! \left(x \right)\\ F_{71}\! \left(x \right) &= -F_{77}\! \left(x \right)+F_{72}\! \left(x \right)\\ F_{72}\! \left(x \right) &= \frac{F_{73}\! \left(x \right)}{F_{15}\! \left(x \right)}\\ F_{73}\! \left(x \right) &= F_{74}\! \left(x \right)\\ F_{74}\! \left(x \right) &= -F_{75}\! \left(x \right)+F_{53}\! \left(x \right)\\ F_{75}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{76}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{26}\! \left(x \right)\\ F_{77}\! \left(x \right) &= F_{78}\! \left(x \right)\\ F_{78}\! \left(x \right) &= F_{15}\! \left(x \right) F_{53}\! \left(x \right) F_{59}\! \left(x \right)\\ F_{79}\! \left(x \right) &= -F_{151}\! \left(x \right)+F_{80}\! \left(x \right)\\ F_{80}\! \left(x \right) &= F_{145}\! \left(x \right)+F_{81}\! \left(x \right)\\ F_{81}\! \left(x \right) &= F_{82}\! \left(x \right)+F_{84}\! \left(x \right)\\ F_{82}\! \left(x \right) &= F_{83}\! \left(x \right)\\ F_{83}\! \left(x \right) &= F_{15}\! \left(x \right) F_{77}\! \left(x \right)\\ F_{84}\! \left(x \right) &= -F_{144}\! \left(x \right)+F_{85}\! \left(x \right)\\ F_{85}\! \left(x \right) &= F_{86}\! \left(x \right)\\ F_{86}\! \left(x \right) &= F_{15}\! \left(x \right) F_{87}\! \left(x \right)\\ F_{87}\! \left(x \right) &= F_{134}\! \left(x \right)+F_{88}\! \left(x \right)\\ F_{88}\! \left(x \right) &= F_{52}\! \left(x \right) F_{89}\! \left(x \right)\\ F_{89}\! \left(x \right) &= \frac{F_{90}\! \left(x \right)}{F_{125}\! \left(x \right)}\\ F_{90}\! \left(x \right) &= -F_{123}\! \left(x \right)+F_{91}\! \left(x \right)\\ F_{91}\! \left(x \right) &= \frac{F_{92}\! \left(x \right)}{F_{15}\! \left(x \right)}\\ F_{92}\! \left(x \right) &= F_{93}\! \left(x \right)\\ F_{93}\! \left(x \right) &= -F_{97}\! \left(x \right)+F_{94}\! \left(x \right)\\ F_{94}\! \left(x \right) &= \frac{F_{95}\! \left(x \right)}{F_{15}\! \left(x \right)}\\ F_{95}\! \left(x \right) &= F_{96}\! \left(x \right)\\ F_{96}\! \left(x \right) &= -F_{2}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{97}\! \left(x \right) &= F_{106}\! \left(x \right)+F_{98}\! \left(x \right)\\ F_{98}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{99}\! \left(x \right)\\ F_{99}\! \left(x \right) &= F_{0}\! \left(x \right) F_{2}\! \left(x \right)\\ F_{100}\! \left(x \right) &= F_{101}\! \left(x \right)\\ F_{101}\! \left(x \right) &= F_{102}\! \left(x \right) F_{15}\! \left(x \right)\\ F_{102}\! \left(x \right) &= F_{103}\! \left(x \right)+F_{104}\! \left(x \right)\\ F_{103}\! \left(x \right) &= F_{4}\! \left(x \right) F_{41}\! \left(x \right)\\ F_{104}\! \left(x \right) &= F_{105}\! \left(x \right)\\ F_{105}\! \left(x \right) &= F_{15}\! \left(x \right) F_{35}\! \left(x \right) F_{53}\! \left(x \right)\\ F_{106}\! \left(x \right) &= F_{107}\! \left(x \right)\\ F_{107}\! \left(x \right) &= F_{108}\! \left(x \right) F_{15}\! \left(x \right)\\ F_{108}\! \left(x \right) &= F_{109}\! \left(x \right)+F_{112}\! \left(x \right)\\ F_{109}\! \left(x \right) &= F_{110}\! \left(x \right) F_{89}\! \left(x \right)\\ F_{110}\! \left(x \right) &= \frac{F_{111}\! \left(x \right)}{F_{15}\! \left(x \right)}\\ F_{111}\! \left(x \right) &= F_{12}\! \left(x \right)\\ F_{112}\! \left(x \right) &= F_{113}\! \left(x \right)+F_{120}\! \left(x \right)\\ F_{113}\! \left(x \right) &= F_{114}\! \left(x \right)\\ F_{114}\! \left(x \right) &= F_{115}\! \left(x \right) F_{116}\! \left(x \right) F_{15}\! \left(x \right)\\ F_{115}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{13}\! \left(x \right)\\ F_{116}\! \left(x \right) &= F_{117}\! \left(x \right)+F_{118}\! \left(x \right)\\ F_{117}\! \left(x \right) &= F_{59}\! \left(x \right) F_{89}\! \left(x \right)\\ F_{118}\! \left(x \right) &= F_{119}\! \left(x \right)\\ F_{119}\! \left(x \right) &= F_{15}\! \left(x \right) F_{35}\! \left(x \right) F_{59}\! \left(x \right)\\ F_{120}\! \left(x \right) &= F_{121}\! \left(x \right)\\ F_{121}\! \left(x \right) &= F_{116}\! \left(x \right) F_{122}\! \left(x \right) F_{15}\! \left(x \right)\\ F_{122}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{74}\! \left(x \right)\\ F_{123}\! \left(x \right) &= F_{124}\! \left(x \right)\\ F_{124}\! \left(x \right) &= F_{116}\! \left(x \right) F_{15}\! \left(x \right)\\ F_{125}\! \left(x \right) &= \frac{F_{126}\! \left(x \right)}{F_{0}\! \left(x \right)}\\ F_{126}\! \left(x \right) &= -F_{129}\! \left(x \right)+F_{127}\! \left(x \right)\\ F_{127}\! \left(x \right) &= \frac{F_{128}\! \left(x \right)}{F_{15}\! \left(x \right)}\\ F_{128}\! \left(x \right) &= F_{20}\! \left(x \right)\\ F_{129}\! \left(x \right) &= -F_{133}\! \left(x \right)+F_{130}\! \left(x \right)\\ F_{130}\! \left(x \right) &= \frac{F_{131}\! \left(x \right)}{F_{15}\! \left(x \right)}\\ F_{131}\! \left(x \right) &= F_{132}\! \left(x \right)\\ F_{132}\! \left(x \right) &= -F_{2}\! \left(x \right)+F_{18}\! \left(x \right)\\ F_{133}\! \left(x \right) &= F_{125}\! \left(x \right) F_{2}\! \left(x \right)\\ F_{134}\! \left(x \right) &= F_{135}\! \left(x \right)\\ F_{135}\! \left(x \right) &= -F_{143}\! \left(x \right)+F_{136}\! \left(x \right)\\ F_{136}\! \left(x \right) &= \frac{F_{137}\! \left(x \right)}{F_{15}\! \left(x \right)}\\ F_{137}\! \left(x \right) &= F_{138}\! \left(x \right)\\ F_{138}\! \left(x \right) &= F_{139}\! \left(x \right) F_{15}\! \left(x \right) F_{59}\! \left(x \right)\\ F_{139}\! \left(x \right) &= F_{140}\! \left(x \right)+F_{141}\! \left(x \right)\\ F_{140}\! \left(x \right) &= F_{53}\! \left(x \right) F_{59}\! \left(x \right) F_{89}\! \left(x \right)\\ F_{141}\! \left(x \right) &= F_{142}\! \left(x \right)\\ F_{142}\! \left(x \right) &= F_{116}\! \left(x \right) F_{15}\! \left(x \right) F_{53}\! \left(x \right)\\ F_{143}\! \left(x \right) &= F_{51}\! \left(x \right) F_{89}\! \left(x \right)\\ F_{144}\! \left(x \right) &= F_{50}\! \left(x \right)\\ F_{145}\! \left(x \right) &= F_{146}\! \left(x \right)\\ F_{146}\! \left(x \right) &= -F_{82}\! \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_{15}\! \left(x \right)\\ F_{149}\! \left(x \right) &= F_{150}\! \left(x \right)\\ F_{150}\! \left(x \right) &= F_{15}\! \left(x \right) F_{53}\! \left(x \right) F_{59}\! \left(x \right) F_{89}\! \left(x \right)\\ F_{151}\! \left(x \right) &= F_{152}\! \left(x \right)\\ F_{152}\! \left(x \right) &= F_{134}\! \left(x \right) F_{15}\! \left(x \right)\\ F_{153}\! \left(x \right) &= F_{154}\! \left(x \right)\\ F_{154}\! \left(x \right) &= F_{15}\! \left(x \right) F_{155}\! \left(x \right)\\ F_{155}\! \left(x \right) &= F_{156}\! \left(x \right)+F_{157}\! \left(x \right)\\ F_{156}\! \left(x \right) &= F_{116}\! \left(x \right) F_{71}\! \left(x \right)\\ F_{157}\! \left(x \right) &= \frac{F_{158}\! \left(x \right)}{F_{15}\! \left(x \right)}\\ F_{158}\! \left(x \right) &= F_{152}\! \left(x \right)\\ F_{159}\! \left(x \right) &= F_{82}\! \left(x \right)\\ \end{align*}\)

This specification was found using the strategy pack "Point Placements Req Corrob" and has 160 rules.

Finding the specification took 11007 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_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{15}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{45}\! \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_{42}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{12}\! \left(x \right)\\ F_{11}\! \left(x \right) &= x^{2} F_{11} \left(x \right)^{2}+2 x^{2} F_{11}\! \left(x \right)-2 x F_{11} \left(x \right)^{2}+x^{2}-3 x F_{11}\! \left(x \right)-x +2 F_{11}\! \left(x \right)\\ F_{12}\! \left(x \right) &= -F_{2}\! \left(x \right)+F_{13}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{15}\! \left(x \right) &= x\\ F_{16}\! \left(x \right) &= \frac{F_{17}\! \left(x \right)}{F_{15}\! \left(x \right)}\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\ F_{18}\! \left(x \right) &= -F_{26}\! \left(x \right)+F_{19}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{20}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{15}\! \left(x \right) F_{22}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{37}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{24}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{2}\! \left(x \right) F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= x^{2} F_{26} \left(x \right)^{2}-2 x F_{26} \left(x \right)^{2}+F_{26}\! \left(x \right) x +2 F_{26}\! \left(x \right)-1\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{15}\! \left(x \right) F_{29}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{33}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{18}\! \left(x \right) F_{31}\! \left(x \right)\\ F_{31}\! \left(x \right) &= \frac{F_{32}\! \left(x \right)}{F_{15}\! \left(x \right)}\\ F_{32}\! \left(x \right) &= F_{2}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{15}\! \left(x \right) F_{16}\! \left(x \right) F_{35}\! \left(x \right)\\ F_{35}\! \left(x \right) &= \frac{F_{36}\! \left(x \right)}{F_{15}\! \left(x \right) F_{26}\! \left(x \right)}\\ F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)\\ F_{37}\! \left(x \right) &= -F_{24}\! \left(x \right)+F_{38}\! \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_{40}\! \left(x \right) &= -F_{2}\! \left(x \right)+F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= -F_{0}\! \left(x \right)+F_{31}\! \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_{10}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{62}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{15}\! \left(x \right) F_{49}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{15}\! \left(x \right) F_{51}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)+F_{57}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)+F_{55}\! \left(x \right)\\ F_{53}\! \left(x \right) &= \frac{F_{54}\! \left(x \right)}{F_{15}\! \left(x \right)}\\ F_{54}\! \left(x \right) &= F_{5}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{15}\! \left(x \right) F_{51}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{15}\! \left(x \right) F_{53}\! \left(x \right) F_{59}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{60}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{61}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{15}\! \left(x \right) F_{59}\! \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_{159}\! \left(x \right)+F_{65}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{153}\! \left(x \right)+F_{66}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{67}\! \left(x \right)+F_{79}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{2}\! \left(x \right) F_{68}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{69}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{70}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{15}\! \left(x \right) F_{71}\! \left(x \right)\\ F_{71}\! \left(x \right) &= -F_{77}\! \left(x \right)+F_{72}\! \left(x \right)\\ F_{72}\! \left(x \right) &= \frac{F_{73}\! \left(x \right)}{F_{15}\! \left(x \right)}\\ F_{73}\! \left(x \right) &= F_{74}\! \left(x \right)\\ F_{74}\! \left(x \right) &= -F_{75}\! \left(x \right)+F_{53}\! \left(x \right)\\ F_{75}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{76}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{26}\! \left(x \right)\\ F_{77}\! \left(x \right) &= F_{78}\! \left(x \right)\\ F_{78}\! \left(x \right) &= F_{15}\! \left(x \right) F_{53}\! \left(x \right) F_{59}\! \left(x \right)\\ F_{79}\! \left(x \right) &= -F_{151}\! \left(x \right)+F_{80}\! \left(x \right)\\ F_{80}\! \left(x \right) &= F_{145}\! \left(x \right)+F_{81}\! \left(x \right)\\ F_{81}\! \left(x \right) &= F_{82}\! \left(x \right)+F_{84}\! \left(x \right)\\ F_{82}\! \left(x \right) &= F_{83}\! \left(x \right)\\ F_{83}\! \left(x \right) &= F_{15}\! \left(x \right) F_{77}\! \left(x \right)\\ F_{84}\! \left(x \right) &= -F_{144}\! \left(x \right)+F_{85}\! \left(x \right)\\ F_{85}\! \left(x \right) &= F_{86}\! \left(x \right)\\ F_{86}\! \left(x \right) &= F_{15}\! \left(x \right) F_{87}\! \left(x \right)\\ F_{87}\! \left(x \right) &= F_{134}\! \left(x \right)+F_{88}\! \left(x \right)\\ F_{88}\! \left(x \right) &= F_{52}\! \left(x \right) F_{89}\! \left(x \right)\\ F_{89}\! \left(x \right) &= \frac{F_{90}\! \left(x \right)}{F_{125}\! \left(x \right)}\\ F_{90}\! \left(x \right) &= -F_{123}\! \left(x \right)+F_{91}\! \left(x \right)\\ F_{91}\! \left(x \right) &= \frac{F_{92}\! \left(x \right)}{F_{15}\! \left(x \right)}\\ F_{92}\! \left(x \right) &= F_{93}\! \left(x \right)\\ F_{93}\! \left(x \right) &= -F_{97}\! \left(x \right)+F_{94}\! \left(x \right)\\ F_{94}\! \left(x \right) &= \frac{F_{95}\! \left(x \right)}{F_{15}\! \left(x \right)}\\ F_{95}\! \left(x \right) &= F_{96}\! \left(x \right)\\ F_{96}\! \left(x \right) &= -F_{2}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{97}\! \left(x \right) &= F_{106}\! \left(x \right)+F_{98}\! \left(x \right)\\ F_{98}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{99}\! \left(x \right)\\ F_{99}\! \left(x \right) &= F_{0}\! \left(x \right) F_{2}\! \left(x \right)\\ F_{100}\! \left(x \right) &= F_{101}\! \left(x \right)\\ F_{101}\! \left(x \right) &= F_{102}\! \left(x \right) F_{15}\! \left(x \right)\\ F_{102}\! \left(x \right) &= F_{103}\! \left(x \right)+F_{104}\! \left(x \right)\\ F_{103}\! \left(x \right) &= F_{4}\! \left(x \right) F_{41}\! \left(x \right)\\ F_{104}\! \left(x \right) &= F_{105}\! \left(x \right)\\ F_{105}\! \left(x \right) &= F_{15}\! \left(x \right) F_{35}\! \left(x \right) F_{53}\! \left(x \right)\\ F_{106}\! \left(x \right) &= F_{107}\! \left(x \right)\\ F_{107}\! \left(x \right) &= F_{108}\! \left(x \right) F_{15}\! \left(x \right)\\ F_{108}\! \left(x \right) &= F_{109}\! \left(x \right)+F_{112}\! \left(x \right)\\ F_{109}\! \left(x \right) &= F_{110}\! \left(x \right) F_{89}\! \left(x \right)\\ F_{110}\! \left(x \right) &= \frac{F_{111}\! \left(x \right)}{F_{15}\! \left(x \right)}\\ F_{111}\! \left(x \right) &= F_{12}\! \left(x \right)\\ F_{112}\! \left(x \right) &= F_{113}\! \left(x \right)+F_{120}\! \left(x \right)\\ F_{113}\! \left(x \right) &= F_{114}\! \left(x \right)\\ F_{114}\! \left(x \right) &= F_{115}\! \left(x \right) F_{116}\! \left(x \right) F_{15}\! \left(x \right)\\ F_{115}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{13}\! \left(x \right)\\ F_{116}\! \left(x \right) &= F_{117}\! \left(x \right)+F_{118}\! \left(x \right)\\ F_{117}\! \left(x \right) &= F_{59}\! \left(x \right) F_{89}\! \left(x \right)\\ F_{118}\! \left(x \right) &= F_{119}\! \left(x \right)\\ F_{119}\! \left(x \right) &= F_{15}\! \left(x \right) F_{35}\! \left(x \right) F_{59}\! \left(x \right)\\ F_{120}\! \left(x \right) &= F_{121}\! \left(x \right)\\ F_{121}\! \left(x \right) &= F_{116}\! \left(x \right) F_{122}\! \left(x \right) F_{15}\! \left(x \right)\\ F_{122}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{74}\! \left(x \right)\\ F_{123}\! \left(x \right) &= F_{124}\! \left(x \right)\\ F_{124}\! \left(x \right) &= F_{116}\! \left(x \right) F_{15}\! \left(x \right)\\ F_{125}\! \left(x \right) &= \frac{F_{126}\! \left(x \right)}{F_{0}\! \left(x \right)}\\ F_{126}\! \left(x \right) &= -F_{129}\! \left(x \right)+F_{127}\! \left(x \right)\\ F_{127}\! \left(x \right) &= \frac{F_{128}\! \left(x \right)}{F_{15}\! \left(x \right)}\\ F_{128}\! \left(x \right) &= F_{20}\! \left(x \right)\\ F_{129}\! \left(x \right) &= -F_{133}\! \left(x \right)+F_{130}\! \left(x \right)\\ F_{130}\! \left(x \right) &= \frac{F_{131}\! \left(x \right)}{F_{15}\! \left(x \right)}\\ F_{131}\! \left(x \right) &= F_{132}\! \left(x \right)\\ F_{132}\! \left(x \right) &= -F_{2}\! \left(x \right)+F_{18}\! \left(x \right)\\ F_{133}\! \left(x \right) &= F_{125}\! \left(x \right) F_{2}\! \left(x \right)\\ F_{134}\! \left(x \right) &= F_{135}\! \left(x \right)\\ F_{135}\! \left(x \right) &= -F_{143}\! \left(x \right)+F_{136}\! \left(x \right)\\ F_{136}\! \left(x \right) &= \frac{F_{137}\! \left(x \right)}{F_{15}\! \left(x \right)}\\ F_{137}\! \left(x \right) &= F_{138}\! \left(x \right)\\ F_{138}\! \left(x \right) &= F_{139}\! \left(x \right) F_{15}\! \left(x \right) F_{59}\! \left(x \right)\\ F_{139}\! \left(x \right) &= F_{140}\! \left(x \right)+F_{141}\! \left(x \right)\\ F_{140}\! \left(x \right) &= F_{53}\! \left(x \right) F_{59}\! \left(x \right) F_{89}\! \left(x \right)\\ F_{141}\! \left(x \right) &= F_{142}\! \left(x \right)\\ F_{142}\! \left(x \right) &= F_{116}\! \left(x \right) F_{15}\! \left(x \right) F_{53}\! \left(x \right)\\ F_{143}\! \left(x \right) &= F_{51}\! \left(x \right) F_{89}\! \left(x \right)\\ F_{144}\! \left(x \right) &= F_{50}\! \left(x \right)\\ F_{145}\! \left(x \right) &= F_{146}\! \left(x \right)\\ F_{146}\! \left(x \right) &= -F_{82}\! \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_{15}\! \left(x \right)\\ F_{149}\! \left(x \right) &= F_{150}\! \left(x \right)\\ F_{150}\! \left(x \right) &= F_{15}\! \left(x \right) F_{53}\! \left(x \right) F_{59}\! \left(x \right) F_{89}\! \left(x \right)\\ F_{151}\! \left(x \right) &= F_{152}\! \left(x \right)\\ F_{152}\! \left(x \right) &= F_{134}\! \left(x \right) F_{15}\! \left(x \right)\\ F_{153}\! \left(x \right) &= F_{154}\! \left(x \right)\\ F_{154}\! \left(x \right) &= F_{15}\! \left(x \right) F_{155}\! \left(x \right)\\ F_{155}\! \left(x \right) &= F_{156}\! \left(x \right)+F_{157}\! \left(x \right)\\ F_{156}\! \left(x \right) &= F_{116}\! \left(x \right) F_{71}\! \left(x \right)\\ F_{157}\! \left(x \right) &= \frac{F_{158}\! \left(x \right)}{F_{15}\! \left(x \right)}\\ F_{158}\! \left(x \right) &= F_{152}\! \left(x \right)\\ F_{159}\! \left(x \right) &= F_{82}\! \left(x \right)\\ \end{align*}\)

This specification was found using the strategy pack "Point And Row Placements Tracked Fusion Tracked Component Fusion Symmetries" and has 94 rules.

Finding the specification took 660 seconds.

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This specification was found using the strategy pack "Point Placements Req Corrob" and has 161 rules.

Finding the specification took 31059 seconds.

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Av(12354, 12453, 12543, 21354, 21453, 21543, 31254)

This specification was found using the strategy pack "Point Placements Req Corrob" and has 161 rules.

Proof Tree

System of Equations

This specification was found using the strategy pack "Point Placements Req Corrob" and has 160 rules.

Proof Tree

System of Equations

This specification was found using the strategy pack "Point Placements Req Corrob" and has 160 rules.

Proof Tree

System of Equations

This specification was found using the strategy pack "Point And Row Placements Tracked Fusion Tracked Component Fusion Symmetries" and has 94 rules.

Proof Tree

System of Equations

This specification was found using the strategy pack "Point Placements Req Corrob" and has 161 rules.

Proof Tree

System of Equations