Av(13425, 13452, 13542, 31425, 31452, 31542, 34125, 34152, 34512, 35142, 35412)
Generating Function
\(\displaystyle \frac{\left(-10 x^{4}+38 x^{3}-48 x^{2}+19 x -2\right) \sqrt{5 x^{2}-6 x +1}+8 x^{6}-70 x^{5}+248 x^{4}-404 x^{3}+303 x^{2}-97 x +10}{16 x^{6}-112 x^{5}+304 x^{4}-402 x^{3}+266 x^{2}-80 x +8}\)
Counting Sequence
1, 1, 2, 6, 24, 109, 522, 2565, 12796, 64493, 327498, 1672550, 8579582, 44162889, 227951798, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(4 x^{2}-6 x +1\right) \left(x -1\right)^{2} \left(-2+x \right)^{2} \left(2 x -1\right)^{2} F \left(x
\right)^{2}-\left(x -1\right) \left(2 x -1\right) \left(8 x^{6}-70 x^{5}+248 x^{4}-404 x^{3}+303 x^{2}-97 x +10\right) F \! \left(x \right)+4 x^{8}-48 x^{7}+218 x^{6}-580 x^{5}+875 x^{4}-733 x^{3}+330 x^{2}-73 x +6 = 0\)
Recurrence
\(\displaystyle a{\left(0 \right)} = 1\)
\(\displaystyle a{\left(1 \right)} = 1\)
\(\displaystyle a{\left(2 \right)} = 2\)
\(\displaystyle a{\left(3 \right)} = 6\)
\(\displaystyle a{\left(4 \right)} = 24\)
\(\displaystyle a{\left(5 \right)} = 109\)
\(\displaystyle a{\left(6 \right)} = 522\)
\(\displaystyle a{\left(7 \right)} = 2565\)
\(\displaystyle a{\left(8 \right)} = 12796\)
\(\displaystyle a{\left(9 \right)} = 64493\)
\(\displaystyle a{\left(10 \right)} = 327498\)
\(\displaystyle a{\left(11 \right)} = 1672550\)
\(\displaystyle a{\left(n + 10 \right)} = - \frac{100 \left(n + 1\right) a{\left(n \right)}}{n + 10} + \frac{20 \left(45 n + 77\right) a{\left(n + 1 \right)}}{n + 10} + \frac{\left(48 n + 427\right) a{\left(n + 9 \right)}}{2 \left(n + 10\right)} - \frac{48 \left(72 n + 179\right) a{\left(n + 2 \right)}}{n + 10} - \frac{3 \left(317 n + 2485\right) a{\left(n + 8 \right)}}{4 \left(n + 10\right)} + \frac{5 \left(507 n + 3464\right) a{\left(n + 7 \right)}}{2 \left(n + 10\right)} + \frac{3 \left(5203 n + 25911\right) a{\left(n + 5 \right)}}{2 \left(n + 10\right)} + \frac{\left(14757 n + 48605\right) a{\left(n + 3 \right)}}{2 \left(n + 10\right)} - \frac{\left(16001 n + 94079\right) a{\left(n + 6 \right)}}{4 \left(n + 10\right)} - \frac{\left(19159 n + 78991\right) a{\left(n + 4 \right)}}{2 \left(n + 10\right)} + \frac{3}{2 \left(n + 10\right)}, \quad n \geq 12\)
\(\displaystyle a{\left(1 \right)} = 1\)
\(\displaystyle a{\left(2 \right)} = 2\)
\(\displaystyle a{\left(3 \right)} = 6\)
\(\displaystyle a{\left(4 \right)} = 24\)
\(\displaystyle a{\left(5 \right)} = 109\)
\(\displaystyle a{\left(6 \right)} = 522\)
\(\displaystyle a{\left(7 \right)} = 2565\)
\(\displaystyle a{\left(8 \right)} = 12796\)
\(\displaystyle a{\left(9 \right)} = 64493\)
\(\displaystyle a{\left(10 \right)} = 327498\)
\(\displaystyle a{\left(11 \right)} = 1672550\)
\(\displaystyle a{\left(n + 10 \right)} = - \frac{100 \left(n + 1\right) a{\left(n \right)}}{n + 10} + \frac{20 \left(45 n + 77\right) a{\left(n + 1 \right)}}{n + 10} + \frac{\left(48 n + 427\right) a{\left(n + 9 \right)}}{2 \left(n + 10\right)} - \frac{48 \left(72 n + 179\right) a{\left(n + 2 \right)}}{n + 10} - \frac{3 \left(317 n + 2485\right) a{\left(n + 8 \right)}}{4 \left(n + 10\right)} + \frac{5 \left(507 n + 3464\right) a{\left(n + 7 \right)}}{2 \left(n + 10\right)} + \frac{3 \left(5203 n + 25911\right) a{\left(n + 5 \right)}}{2 \left(n + 10\right)} + \frac{\left(14757 n + 48605\right) a{\left(n + 3 \right)}}{2 \left(n + 10\right)} - \frac{\left(16001 n + 94079\right) a{\left(n + 6 \right)}}{4 \left(n + 10\right)} - \frac{\left(19159 n + 78991\right) a{\left(n + 4 \right)}}{2 \left(n + 10\right)} + \frac{3}{2 \left(n + 10\right)}, \quad n \geq 12\)
This specification was found using the strategy pack "Point Placements Req Corrob" and has 97 rules.
Finding the specification took 1662 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_{21}\! \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_{21}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{8}\! \left(x \right)+F_{84}\! \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_{10}\! \left(x \right) &= F_{11}\! \left(x \right) F_{21}\! \left(x \right) F_{40}\! \left(x \right)\\
F_{11}\! \left(x \right) &= \frac{F_{12}\! \left(x \right)}{F_{39}\! \left(x \right)}\\
F_{12}\! \left(x \right) &= -F_{23}\! \left(x \right)+F_{13}\! \left(x \right)\\
F_{13}\! \left(x \right) &= \frac{F_{14}\! \left(x \right)}{F_{21}\! \left(x \right)}\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{16}\! \left(x \right) &= x^{2} F_{16} \left(x \right)^{2}+2 x^{2} F_{16}\! \left(x \right)-2 x F_{16} \left(x \right)^{2}+x^{2}-3 x F_{16}\! \left(x \right)-x +2 F_{16}\! \left(x \right)\\
F_{17}\! \left(x \right) &= -F_{2}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= -F_{22}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= \frac{F_{20}\! \left(x \right)}{F_{21}\! \left(x \right)}\\
F_{20}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{21}\! \left(x \right) &= x\\
F_{22}\! \left(x \right) &= x^{2} F_{22} \left(x \right)^{2}-2 x F_{22} \left(x \right)^{2}+F_{22}\! \left(x \right) x +2 F_{22}\! \left(x \right)-1\\
F_{23}\! \left(x \right) &= -F_{26}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= \frac{F_{25}\! \left(x \right)}{F_{21}\! \left(x \right)}\\
F_{25}\! \left(x \right) &= F_{17}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{11}\! \left(x \right) F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{21}\! \left(x \right) F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{21}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{34}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{33}\! \left(x \right) &= 0\\
F_{34}\! \left(x \right) &= F_{21}\! \left(x \right) F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{21}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{32}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{21}\! \left(x \right) F_{27}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{40}\! \left(x \right) &= \frac{F_{41}\! \left(x \right)}{F_{21}\! \left(x \right) F_{22}\! \left(x \right)}\\
F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{21}\! \left(x \right) F_{22}\! \left(x \right) F_{43}\! \left(x \right) F_{46}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{21}\! \left(x \right) F_{43}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{50}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{21}\! \left(x \right) F_{40}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{21}\! \left(x \right) F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)+F_{82}\! \left(x \right)\\
F_{53}\! \left(x \right) &= \frac{F_{54}\! \left(x \right)}{F_{21}\! \left(x \right) F_{79}\! \left(x \right)}\\
F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= -F_{58}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= \frac{F_{57}\! \left(x \right)}{F_{21}\! \left(x \right)}\\
F_{57}\! \left(x \right) &= F_{5}\! \left(x \right)\\
F_{58}\! \left(x \right) &= -F_{59}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{60}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{21}\! \left(x \right) F_{46}\! \left(x \right) F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= \frac{F_{63}\! \left(x \right)}{F_{21}\! \left(x \right)}\\
F_{63}\! \left(x \right) &= F_{64}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{11}\! \left(x \right) F_{21}\! \left(x \right) F_{66}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{67}\! \left(x \right)+F_{71}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{68}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{69}\! \left(x \right)+F_{70}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{21}\! \left(x \right) F_{67}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{21}\! \left(x \right) F_{44}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{72}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{21}\! \left(x \right) F_{73}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{74}\! \left(x \right)+F_{78}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{66}\! \left(x \right) F_{75}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{76}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{77}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{75} \left(x \right)^{2} F_{21}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{66}\! \left(x \right) F_{75}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{80}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{81}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{21}\! \left(x \right) F_{75}\! \left(x \right) F_{79}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{83}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{50}\! \left(x \right) F_{75}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{85}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{21}\! \left(x \right) F_{46}\! \left(x \right) F_{86}\! \left(x \right)\\
F_{86}\! \left(x \right) &= \frac{F_{87}\! \left(x \right)}{F_{21}\! \left(x \right)}\\
F_{87}\! \left(x \right) &= F_{88}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{64}\! \left(x \right)+F_{89}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{90}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{91}\! \left(x \right)\\
F_{91}\! \left(x \right) &= F_{21}\! \left(x \right) F_{92}\! \left(x \right)\\
F_{92}\! \left(x \right) &= F_{88}\! \left(x \right)+F_{93}\! \left(x \right)\\
F_{93}\! \left(x \right) &= F_{94}\! \left(x \right)+F_{95}\! \left(x \right)\\
F_{94}\! \left(x \right) &= F_{21}\! \left(x \right) F_{22}\! \left(x \right)\\
F_{95}\! \left(x \right) &= F_{96}\! \left(x \right)\\
F_{96}\! \left(x \right) &= F_{21}\! \left(x \right) F_{75}\! \left(x \right) F_{92}\! \left(x \right)\\
\end{align*}\)
This specification was found using the strategy pack "Point Placements Req Corrob" and has 97 rules.
Finding the specification took 1662 seconds.
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Copy 97 equations to clipboard:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{21}\! \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_{21}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{8}\! \left(x \right)+F_{84}\! \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_{10}\! \left(x \right) &= F_{11}\! \left(x \right) F_{21}\! \left(x \right) F_{40}\! \left(x \right)\\
F_{11}\! \left(x \right) &= \frac{F_{12}\! \left(x \right)}{F_{39}\! \left(x \right)}\\
F_{12}\! \left(x \right) &= -F_{23}\! \left(x \right)+F_{13}\! \left(x \right)\\
F_{13}\! \left(x \right) &= \frac{F_{14}\! \left(x \right)}{F_{21}\! \left(x \right)}\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{16}\! \left(x \right) &= x^{2} F_{16} \left(x \right)^{2}+2 x^{2} F_{16}\! \left(x \right)-2 x F_{16} \left(x \right)^{2}+x^{2}-3 x F_{16}\! \left(x \right)-x +2 F_{16}\! \left(x \right)\\
F_{17}\! \left(x \right) &= -F_{2}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= -F_{22}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= \frac{F_{20}\! \left(x \right)}{F_{21}\! \left(x \right)}\\
F_{20}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{21}\! \left(x \right) &= x\\
F_{22}\! \left(x \right) &= x^{2} F_{22} \left(x \right)^{2}-2 x F_{22} \left(x \right)^{2}+F_{22}\! \left(x \right) x +2 F_{22}\! \left(x \right)-1\\
F_{23}\! \left(x \right) &= -F_{26}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= \frac{F_{25}\! \left(x \right)}{F_{21}\! \left(x \right)}\\
F_{25}\! \left(x \right) &= F_{17}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{11}\! \left(x \right) F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{21}\! \left(x \right) F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{21}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{34}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{33}\! \left(x \right) &= 0\\
F_{34}\! \left(x \right) &= F_{21}\! \left(x \right) F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{21}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{32}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{21}\! \left(x \right) F_{27}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{40}\! \left(x \right) &= \frac{F_{41}\! \left(x \right)}{F_{21}\! \left(x \right) F_{22}\! \left(x \right)}\\
F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{21}\! \left(x \right) F_{22}\! \left(x \right) F_{43}\! \left(x \right) F_{46}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{21}\! \left(x \right) F_{43}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{50}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{21}\! \left(x \right) F_{40}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{21}\! \left(x \right) F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)+F_{82}\! \left(x \right)\\
F_{53}\! \left(x \right) &= \frac{F_{54}\! \left(x \right)}{F_{21}\! \left(x \right) F_{79}\! \left(x \right)}\\
F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= -F_{58}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= \frac{F_{57}\! \left(x \right)}{F_{21}\! \left(x \right)}\\
F_{57}\! \left(x \right) &= F_{5}\! \left(x \right)\\
F_{58}\! \left(x \right) &= -F_{59}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{60}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{21}\! \left(x \right) F_{46}\! \left(x \right) F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= \frac{F_{63}\! \left(x \right)}{F_{21}\! \left(x \right)}\\
F_{63}\! \left(x \right) &= F_{64}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{11}\! \left(x \right) F_{21}\! \left(x \right) F_{66}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{67}\! \left(x \right)+F_{71}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{68}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{69}\! \left(x \right)+F_{70}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{21}\! \left(x \right) F_{67}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{21}\! \left(x \right) F_{44}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{72}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{21}\! \left(x \right) F_{73}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{74}\! \left(x \right)+F_{78}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{66}\! \left(x \right) F_{75}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{76}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{77}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{75} \left(x \right)^{2} F_{21}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{66}\! \left(x \right) F_{75}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{80}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{81}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{21}\! \left(x \right) F_{75}\! \left(x \right) F_{79}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{83}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{50}\! \left(x \right) F_{75}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{85}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{21}\! \left(x \right) F_{46}\! \left(x \right) F_{86}\! \left(x \right)\\
F_{86}\! \left(x \right) &= \frac{F_{87}\! \left(x \right)}{F_{21}\! \left(x \right)}\\
F_{87}\! \left(x \right) &= F_{88}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{64}\! \left(x \right)+F_{89}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{90}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{91}\! \left(x \right)\\
F_{91}\! \left(x \right) &= F_{21}\! \left(x \right) F_{92}\! \left(x \right)\\
F_{92}\! \left(x \right) &= F_{88}\! \left(x \right)+F_{93}\! \left(x \right)\\
F_{93}\! \left(x \right) &= F_{94}\! \left(x \right)+F_{95}\! \left(x \right)\\
F_{94}\! \left(x \right) &= F_{21}\! \left(x \right) F_{22}\! \left(x \right)\\
F_{95}\! \left(x \right) &= F_{96}\! \left(x \right)\\
F_{96}\! \left(x \right) &= F_{21}\! \left(x \right) F_{75}\! \left(x \right) F_{92}\! \left(x \right)\\
\end{align*}\)
This specification was found using the strategy pack "Point Placements Req Corrob" and has 96 rules.
Finding the specification took 929 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_{18}\! \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_{18}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{8}\! \left(x \right)+F_{83}\! \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_{10}\! \left(x \right) &= F_{11}\! \left(x \right) F_{18}\! \left(x \right) F_{38}\! \left(x \right)\\
F_{11}\! \left(x \right) &= \frac{F_{12}\! \left(x \right)}{F_{37}\! \left(x \right)}\\
F_{12}\! \left(x \right) &= -F_{19}\! \left(x \right)+F_{13}\! \left(x \right)\\
F_{13}\! \left(x \right) &= \frac{F_{14}\! \left(x \right)}{F_{18}\! \left(x \right)}\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= -F_{0}\! \left(x \right)+F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= \frac{F_{17}\! \left(x \right)}{F_{18}\! \left(x \right)}\\
F_{17}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{18}\! \left(x \right) &= x\\
F_{19}\! \left(x \right) &= -F_{24}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= \frac{F_{21}\! \left(x \right)}{F_{18}\! \left(x \right)}\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= -F_{23}\! \left(x \right)+F_{15}\! \left(x \right)\\
F_{23}\! \left(x \right) &= x^{2} F_{23} \left(x \right)^{2}+2 x^{2} F_{23}\! \left(x \right)-2 x F_{23} \left(x \right)^{2}+x^{2}-3 x F_{23}\! \left(x \right)-x +2 F_{23}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{11}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{18}\! \left(x \right) F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{32}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{31}\! \left(x \right) &= 0\\
F_{32}\! \left(x \right) &= F_{18}\! \left(x \right) F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{30}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{18}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{38}\! \left(x \right) &= \frac{F_{39}\! \left(x \right)}{F_{18}\! \left(x \right) F_{41}\! \left(x \right)}\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{18}\! \left(x \right) F_{41}\! \left(x \right) F_{42}\! \left(x \right) F_{45}\! \left(x \right)\\
F_{41}\! \left(x \right) &= x^{2} F_{41} \left(x \right)^{2}-2 x F_{41} \left(x \right)^{2}+F_{41}\! \left(x \right) x +2 F_{41}\! \left(x \right)-1\\
F_{42}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{18}\! \left(x \right) F_{42}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{18}\! \left(x \right) F_{38}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{18}\! \left(x \right) F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)+F_{81}\! \left(x \right)\\
F_{52}\! \left(x \right) &= \frac{F_{53}\! \left(x \right)}{F_{18}\! \left(x \right) F_{78}\! \left(x \right)}\\
F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= -F_{57}\! \left(x \right)+F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= \frac{F_{56}\! \left(x \right)}{F_{18}\! \left(x \right)}\\
F_{56}\! \left(x \right) &= F_{5}\! \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_{9}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{18}\! \left(x \right) F_{45}\! \left(x \right) F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= \frac{F_{62}\! \left(x \right)}{F_{18}\! \left(x \right)}\\
F_{62}\! \left(x \right) &= F_{63}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{64}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{11}\! \left(x \right) F_{18}\! \left(x \right) F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{66}\! \left(x \right)+F_{70}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{68}\! \left(x \right)+F_{69}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{18}\! \left(x \right) F_{66}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{18}\! \left(x \right) F_{43}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{71}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{18}\! \left(x \right) F_{72}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{73}\! \left(x \right)+F_{77}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{65}\! \left(x \right) F_{74}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{75}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{76}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{74} \left(x \right)^{2} F_{18}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{65}\! \left(x \right) F_{74}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{79}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{80}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{18}\! \left(x \right) F_{74}\! \left(x \right) F_{78}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{82}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{49}\! \left(x \right) F_{74}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{84}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{18}\! \left(x \right) F_{45}\! \left(x \right) F_{85}\! \left(x \right)\\
F_{85}\! \left(x \right) &= \frac{F_{86}\! \left(x \right)}{F_{18}\! \left(x \right)}\\
F_{86}\! \left(x \right) &= F_{87}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{63}\! \left(x \right)+F_{88}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{89}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{90}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{18}\! \left(x \right) F_{91}\! \left(x \right)\\
F_{91}\! \left(x \right) &= F_{87}\! \left(x \right)+F_{92}\! \left(x \right)\\
F_{92}\! \left(x \right) &= F_{93}\! \left(x \right)+F_{94}\! \left(x \right)\\
F_{93}\! \left(x \right) &= F_{18}\! \left(x \right) F_{41}\! \left(x \right)\\
F_{94}\! \left(x \right) &= F_{95}\! \left(x \right)\\
F_{95}\! \left(x \right) &= F_{18}\! \left(x \right) F_{74}\! \left(x \right) F_{91}\! \left(x \right)\\
\end{align*}\)
This specification was found using the strategy pack "Point Placements Req Corrob" and has 96 rules.
Finding the specification took 929 seconds.
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Copy 96 equations to clipboard:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{18}\! \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_{18}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{8}\! \left(x \right)+F_{83}\! \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_{10}\! \left(x \right) &= F_{11}\! \left(x \right) F_{18}\! \left(x \right) F_{38}\! \left(x \right)\\
F_{11}\! \left(x \right) &= \frac{F_{12}\! \left(x \right)}{F_{37}\! \left(x \right)}\\
F_{12}\! \left(x \right) &= -F_{19}\! \left(x \right)+F_{13}\! \left(x \right)\\
F_{13}\! \left(x \right) &= \frac{F_{14}\! \left(x \right)}{F_{18}\! \left(x \right)}\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= -F_{0}\! \left(x \right)+F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= \frac{F_{17}\! \left(x \right)}{F_{18}\! \left(x \right)}\\
F_{17}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{18}\! \left(x \right) &= x\\
F_{19}\! \left(x \right) &= -F_{24}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= \frac{F_{21}\! \left(x \right)}{F_{18}\! \left(x \right)}\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= -F_{23}\! \left(x \right)+F_{15}\! \left(x \right)\\
F_{23}\! \left(x \right) &= x^{2} F_{23} \left(x \right)^{2}+2 x^{2} F_{23}\! \left(x \right)-2 x F_{23} \left(x \right)^{2}+x^{2}-3 x F_{23}\! \left(x \right)-x +2 F_{23}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{11}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{18}\! \left(x \right) F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{32}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{31}\! \left(x \right) &= 0\\
F_{32}\! \left(x \right) &= F_{18}\! \left(x \right) F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{30}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{18}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{38}\! \left(x \right) &= \frac{F_{39}\! \left(x \right)}{F_{18}\! \left(x \right) F_{41}\! \left(x \right)}\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{18}\! \left(x \right) F_{41}\! \left(x \right) F_{42}\! \left(x \right) F_{45}\! \left(x \right)\\
F_{41}\! \left(x \right) &= x^{2} F_{41} \left(x \right)^{2}-2 x F_{41} \left(x \right)^{2}+F_{41}\! \left(x \right) x +2 F_{41}\! \left(x \right)-1\\
F_{42}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{18}\! \left(x \right) F_{42}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{18}\! \left(x \right) F_{38}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{18}\! \left(x \right) F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)+F_{81}\! \left(x \right)\\
F_{52}\! \left(x \right) &= \frac{F_{53}\! \left(x \right)}{F_{18}\! \left(x \right) F_{78}\! \left(x \right)}\\
F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= -F_{57}\! \left(x \right)+F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= \frac{F_{56}\! \left(x \right)}{F_{18}\! \left(x \right)}\\
F_{56}\! \left(x \right) &= F_{5}\! \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_{9}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{18}\! \left(x \right) F_{45}\! \left(x \right) F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= \frac{F_{62}\! \left(x \right)}{F_{18}\! \left(x \right)}\\
F_{62}\! \left(x \right) &= F_{63}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{64}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{11}\! \left(x \right) F_{18}\! \left(x \right) F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{66}\! \left(x \right)+F_{70}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{68}\! \left(x \right)+F_{69}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{18}\! \left(x \right) F_{66}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{18}\! \left(x \right) F_{43}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{71}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{18}\! \left(x \right) F_{72}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{73}\! \left(x \right)+F_{77}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{65}\! \left(x \right) F_{74}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{75}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{76}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{74} \left(x \right)^{2} F_{18}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{65}\! \left(x \right) F_{74}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{79}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{80}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{18}\! \left(x \right) F_{74}\! \left(x \right) F_{78}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{82}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{49}\! \left(x \right) F_{74}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{84}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{18}\! \left(x \right) F_{45}\! \left(x \right) F_{85}\! \left(x \right)\\
F_{85}\! \left(x \right) &= \frac{F_{86}\! \left(x \right)}{F_{18}\! \left(x \right)}\\
F_{86}\! \left(x \right) &= F_{87}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{63}\! \left(x \right)+F_{88}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{89}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{90}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{18}\! \left(x \right) F_{91}\! \left(x \right)\\
F_{91}\! \left(x \right) &= F_{87}\! \left(x \right)+F_{92}\! \left(x \right)\\
F_{92}\! \left(x \right) &= F_{93}\! \left(x \right)+F_{94}\! \left(x \right)\\
F_{93}\! \left(x \right) &= F_{18}\! \left(x \right) F_{41}\! \left(x \right)\\
F_{94}\! \left(x \right) &= F_{95}\! \left(x \right)\\
F_{95}\! \left(x \right) &= F_{18}\! \left(x \right) F_{74}\! \left(x \right) F_{91}\! \left(x \right)\\
\end{align*}\)
This specification was found using the strategy pack "Point And Col Placements Tracked Fusion Tracked Component Fusion Symmetries" and has 111 rules.
Finding the specification took 1371 seconds.
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Copy 111 equations to clipboard:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{4}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{6}\! \left(x \right)+F_{7}\! \left(x \right)+F_{87}\! \left(x \right)\\
F_{6}\! \left(x \right) &= 0\\
F_{7}\! \left(x \right) &= F_{8}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{8}\! \left(x \right) &= x\\
F_{9}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x \right)+F_{23}\! \left(x \right)+F_{84}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{12}\! \left(x \right)+F_{23}\! \left(x \right)+F_{50}\! \left(x \right)+F_{82}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x , 1\right)\\
F_{14}\! \left(x , y\right) &= -\frac{-F_{15}\! \left(x , y\right) y +F_{15}\! \left(x , 1\right)}{-1+y}\\
F_{15}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{16}\! \left(x , y\right)+F_{18}\! \left(x , y\right)+F_{23}\! \left(x \right)+F_{76}\! \left(x , y\right)\\
F_{16}\! \left(x , y\right) &= F_{17}\! \left(x , y\right) F_{8}\! \left(x \right)\\
F_{17}\! \left(x , y\right) &= -\frac{-F_{15}\! \left(x , y\right) y +F_{15}\! \left(x , 1\right)}{-1+y}\\
F_{18}\! \left(x , y\right) &= y F_{19}\! \left(x , y\right)\\
F_{19}\! \left(x , y\right) &= F_{20}\! \left(x , y\right) F_{8}\! \left(x \right)\\
F_{20}\! \left(x , y\right) &= -\frac{-y F_{21}\! \left(x , y\right)+F_{21}\! \left(x , 1\right)}{-1+y}\\
F_{21}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{18}\! \left(x , y\right)+F_{22}\! \left(x , y\right)+F_{23}\! \left(x \right)+F_{75}\! \left(x , y\right)\\
F_{22}\! \left(x , y\right) &= F_{21}\! \left(x , y\right) F_{8}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{60}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{23}\! \left(x \right)+F_{26}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{25}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{29}\! \left(x \right)+F_{30}\! \left(x \right)+F_{56}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{28}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x , 1\right)\\
F_{32}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{30}\! \left(x \right)+F_{33}\! \left(x , y\right)+F_{34}\! \left(x , y\right)+F_{37}\! \left(x , y\right)+F_{55}\! \left(x , y\right)\\
F_{33}\! \left(x , y\right) &= F_{32}\! \left(x , y\right) F_{8}\! \left(x \right)\\
F_{34}\! \left(x , y\right) &= y F_{35}\! \left(x , y\right)\\
F_{35}\! \left(x , y\right) &= F_{36}\! \left(x , y\right) F_{8}\! \left(x \right)\\
F_{36}\! \left(x , y\right) &= -\frac{-F_{32}\! \left(x , y\right) y +F_{32}\! \left(x , 1\right)}{-1+y}\\
F_{37}\! \left(x , y\right) &= F_{38}\! \left(x , y\right) F_{8}\! \left(x \right)\\
F_{38}\! \left(x , y\right) &= F_{39}\! \left(x , y\right)+F_{54}\! \left(x , y\right)\\
F_{39}\! \left(x , y\right) &= F_{40}\! \left(x , y\right)+F_{45}\! \left(x , y\right)\\
F_{40}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{23}\! \left(x \right)+F_{27}\! \left(x \right)+F_{41}\! \left(x , y\right)+F_{42}\! \left(x , y\right)\\
F_{41}\! \left(x , y\right) &= F_{40}\! \left(x , y\right) F_{8}\! \left(x \right)\\
F_{42}\! \left(x , y\right) &= y F_{43}\! \left(x , y\right)\\
F_{43}\! \left(x , y\right) &= F_{44}\! \left(x , y\right) F_{8}\! \left(x \right)\\
F_{44}\! \left(x , y\right) &= -\frac{-F_{40}\! \left(x , y\right) y +F_{40}\! \left(x , 1\right)}{-1+y}\\
F_{45}\! \left(x , y\right) &= F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= -F_{53}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= \frac{F_{48}\! \left(x \right)}{F_{8}\! \left(x \right)}\\
F_{48}\! \left(x \right) &= -F_{1}\! \left(x \right)-F_{23}\! \left(x \right)-F_{49}\! \left(x \right)-F_{50}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{22}\! \left(x , 1\right)\\
F_{50}\! \left(x \right) &= F_{51}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{52}\! \left(x , 1\right)\\
F_{52}\! \left(x , y\right) &= -\frac{-y F_{21}\! \left(x , y\right)+F_{21}\! \left(x , 1\right)}{-1+y}\\
F_{53}\! \left(x \right) &= F_{40}\! \left(x , 1\right)\\
F_{54}\! \left(x , y\right) &= F_{46}\! \left(x \right)\\
F_{55}\! \left(x , y\right) &= F_{40}\! \left(x , y\right) F_{8}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{57}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{46}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{25}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{25}\! \left(x \right) F_{62}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{63}\! \left(x , 1\right)\\
F_{63}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{64}\! \left(x , y\right)+F_{65}\! \left(x , y\right)+F_{67}\! \left(x , y\right)+F_{69}\! \left(x , y\right)\\
F_{64}\! \left(x , y\right) &= F_{63}\! \left(x , y\right) F_{8}\! \left(x \right)\\
F_{65}\! \left(x , y\right) &= F_{66}\! \left(x , y\right) F_{8}\! \left(x \right)\\
F_{66}\! \left(x , y\right) &= -\frac{-F_{63}\! \left(x , y\right) y +F_{63}\! \left(x , 1\right)}{-1+y}\\
F_{67}\! \left(x , y\right) &= F_{63}\! \left(x , y\right) F_{68}\! \left(x , y\right)\\
F_{68}\! \left(x , y\right) &= y x\\
F_{69}\! \left(x , y\right) &= F_{70}\! \left(x , y\right) F_{8}\! \left(x \right)\\
F_{70}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{71}\! \left(x , y\right)+F_{72}\! \left(x , y\right)+F_{74}\! \left(x , y\right)\\
F_{71}\! \left(x , y\right) &= F_{70}\! \left(x , y\right) F_{8}\! \left(x \right)\\
F_{72}\! \left(x , y\right) &= F_{73}\! \left(x , y\right) F_{8}\! \left(x \right)\\
F_{73}\! \left(x , y\right) &= -\frac{-F_{70}\! \left(x , y\right) y +F_{70}\! \left(x , 1\right)}{-1+y}\\
F_{74}\! \left(x , y\right) &= F_{68}\! \left(x , y\right) F_{70}\! \left(x , y\right)\\
F_{75}\! \left(x , y\right) &= F_{39}\! \left(x , y\right) F_{8}\! \left(x \right)\\
F_{76}\! \left(x , y\right) &= F_{77}\! \left(x , y\right) F_{8}\! \left(x \right)\\
F_{77}\! \left(x , y\right) &= F_{78}\! \left(x , y\right)+F_{81}\! \left(x , y\right)\\
F_{78}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{23}\! \left(x \right)+F_{27}\! \left(x \right)+F_{42}\! \left(x , y\right)+F_{79}\! \left(x , y\right)\\
F_{79}\! \left(x , y\right) &= F_{8}\! \left(x \right) F_{80}\! \left(x , y\right)\\
F_{80}\! \left(x , y\right) &= -\frac{-y F_{78}\! \left(x , y\right)+F_{78}\! \left(x , 1\right)}{-1+y}\\
F_{81}\! \left(x , y\right) &= F_{46}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{8}\! \left(x \right) F_{83}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{77}\! \left(x , 1\right)\\
F_{84}\! \left(x \right) &= F_{8}\! \left(x \right) F_{85}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{30}\! \left(x \right)+F_{56}\! \left(x \right)+F_{59}\! \left(x \right)+F_{86}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{8}\! \left(x \right) F_{83}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{8}\! \left(x \right) F_{88}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{5}\! \left(x \right)+F_{89}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{90}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{8}\! \left(x \right) F_{91}\! \left(x \right)\\
F_{91}\! \left(x \right) &= F_{107}\! \left(x \right)+F_{92}\! \left(x \right)\\
F_{92}\! \left(x \right) &= F_{105}\! \left(x \right)+F_{93}\! \left(x \right)\\
F_{93}\! \left(x \right) &= F_{94}\! \left(x \right)\\
F_{94}\! \left(x \right) &= -F_{95}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{95}\! \left(x \right) &= F_{96}\! \left(x \right)+F_{97}\! \left(x \right)\\
F_{96}\! \left(x \right) &= x^{2} F_{96} \left(x \right)^{2}-2 x F_{96} \left(x \right)^{2}+F_{96}\! \left(x \right) x +2 F_{96}\! \left(x \right)-1\\
F_{97}\! \left(x \right) &= F_{98}\! \left(x \right)\\
F_{98}\! \left(x \right) &= F_{8}\! \left(x \right) F_{95}\! \left(x \right) F_{99}\! \left(x \right)\\
F_{99}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{103}\! \left(x \right)\\
F_{100}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{101}\! \left(x \right)\\
F_{101}\! \left(x \right) &= F_{102}\! \left(x \right)\\
F_{102}\! \left(x \right) &= F_{100}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{103}\! \left(x \right) &= F_{104}\! \left(x \right)\\
F_{104}\! \left(x \right) &= F_{99} \left(x \right)^{2} F_{8}\! \left(x \right)\\
F_{105}\! \left(x \right) &= F_{106}\! \left(x \right)\\
F_{106}\! \left(x \right) &= F_{100}\! \left(x \right) F_{8}\! \left(x \right) F_{93}\! \left(x \right)\\
F_{107}\! \left(x \right) &= F_{108}\! \left(x \right)\\
F_{108}\! \left(x \right) &= F_{100}\! \left(x \right) F_{109}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{109}\! \left(x \right) &= \frac{F_{110}\! \left(x \right)}{F_{8}\! \left(x \right)}\\
F_{110}\! \left(x \right) &= F_{5}\! \left(x \right)\\
\end{align*}\)