Av(1234, 1342, 1423)
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Counting Sequence
1, 1, 2, 6, 21, 79, 310, 1251, 5151, 21536, 91137, 389510, 1678565, 7284975, 31811311, ...
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Implicit Equation for the Generating Function
\(\displaystyle x F \left(x \right)^{4}+\left(-x -1\right) F \left(x \right)^{3}+\left(x^{2}-2 x +3\right) F \left(x \right)^{2}+\left(2 x -3\right) F \! \left(x \right)+1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 79\)
\(\displaystyle a \! \left(6\right) = 310\)
\(\displaystyle a \! \left(7\right) = 1251\)
\(\displaystyle a \! \left(8\right) = 5151\)
\(\displaystyle a \! \left(n +9\right) = \frac{n \left(2 n +3\right) \left(2 n -1\right) a \! \left(n \right)}{\left(n +10\right) \left(n +8\right) \left(2 n +17\right)}+\frac{\left(8 n^{3}+83 n^{2}+145 n +55\right) a \! \left(n +1\right)}{\left(n +10\right) \left(n +8\right) \left(2 n +17\right)}+\frac{\left(1151 n^{3}+8520 n^{2}+22579 n +18546\right) a \! \left(n +2\right)}{12 \left(n +10\right) \left(n +8\right) \left(2 n +17\right)}-\frac{\left(2233 n^{3}+55122 n^{2}+279635 n +395634\right) a \! \left(n +3\right)}{24 \left(n +10\right) \left(n +8\right) \left(2 n +17\right)}+\frac{\left(62879 n^{3}+790140 n^{2}+3299443 n +4570050\right) a \! \left(n +4\right)}{24 \left(n +10\right) \left(n +8\right) \left(2 n +17\right)}-\frac{\left(67265 n^{3}+1013004 n^{2}+5056489 n +8372154\right) a \! \left(n +5\right)}{24 \left(n +10\right) \left(n +8\right) \left(2 n +17\right)}+\frac{\left(7612 n^{3}+136431 n^{2}+810569 n +1597989\right) a \! \left(n +6\right)}{6 \left(n +10\right) \left(n +8\right) \left(2 n +17\right)}-\frac{\left(3640 n^{3}+76029 n^{2}+526883 n +1212402\right) a \! \left(n +7\right)}{12 \left(n +10\right) \left(n +8\right) \left(2 n +17\right)}+\frac{\left(458 n^{3}+10893 n^{2}+85993 n +225450\right) a \! \left(n +8\right)}{12 \left(n +10\right) \left(n +8\right) \left(2 n +17\right)}, \quad n \geq 9\)
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This specification was found using the strategy pack "Point Placements Tracked Fusion" and has 43 rules.

Found on July 23, 2021.

Finding the specification took 57 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_{27}\! \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_{27}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{9}\! \left(x , 1\right)\\ F_{9}\! \left(x , y\right) &= F_{10}\! \left(x , y\right)+F_{14}\! \left(x , y\right)\\ F_{10}\! \left(x , y\right) &= F_{0}\! \left(x \right)+F_{11}\! \left(x , y\right)\\ F_{11}\! \left(x , y\right) &= F_{12}\! \left(x , y\right)\\ F_{12}\! \left(x , y\right) &= F_{13}\! \left(x , y\right) F_{9}\! \left(x , y\right)\\ F_{13}\! \left(x , y\right) &= y x\\ F_{14}\! \left(x , y\right) &= F_{15}\! \left(x , y\right)\\ F_{15}\! \left(x , y\right) &= F_{16}\! \left(x , y\right) F_{27}\! \left(x \right)\\ F_{16}\! \left(x , y\right) &= \frac{y F_{17}\! \left(x , y\right)-F_{17}\! \left(x , 1\right)}{-1+y}\\ F_{17}\! \left(x , y\right) &= F_{18}\! \left(x , y\right)+F_{28}\! \left(x , y\right)\\ F_{18}\! \left(x , y\right) &= F_{19}\! \left(x \right) F_{21}\! \left(x , y\right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x , 1\right)\\ F_{20}\! \left(x , y\right) &= F_{21}\! \left(x , y\right)+F_{24}\! \left(x , y\right)\\ F_{21}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{22}\! \left(x , y\right)\\ F_{22}\! \left(x , y\right) &= F_{23}\! \left(x , y\right)\\ F_{23}\! \left(x , y\right) &= F_{13}\! \left(x , y\right) F_{21}\! \left(x , y\right)\\ F_{24}\! \left(x , y\right) &= F_{25}\! \left(x , y\right)\\ F_{25}\! \left(x , y\right) &= F_{21}\! \left(x , y\right) F_{26}\! \left(x , y\right) F_{27}\! \left(x \right)\\ F_{26}\! \left(x , y\right) &= \frac{y F_{20}\! \left(x , y\right)-F_{20}\! \left(x , 1\right)}{-1+y}\\ F_{27}\! \left(x \right) &= x\\ F_{28}\! \left(x , y\right) &= F_{29}\! \left(x , y\right)\\ F_{29}\! \left(x , y\right) &= F_{19}\! \left(x \right) F_{27}\! \left(x \right) F_{30}\! \left(x , y\right)\\ F_{31}\! \left(x , y\right) &= F_{20}\! \left(x , y\right) F_{27}\! \left(x \right) F_{30}\! \left(x , y\right)\\ F_{31}\! \left(x , y\right) &= F_{32}\! \left(x , y\right)\\ F_{33}\! \left(x , y\right) &= F_{32}\! \left(x , y\right)+F_{41}\! \left(x , y\right)\\ F_{33}\! \left(x , y\right) &= F_{34}\! \left(x , y\right)+F_{9}\! \left(x , y\right)\\ F_{34}\! \left(x , y\right) &= F_{35}\! \left(x , y\right)+F_{39}\! \left(x , y\right)\\ F_{35}\! \left(x , y\right) &= F_{21}\! \left(x , y\right) F_{36}\! \left(x , y\right)\\ F_{20}\! \left(x , y\right) &= F_{36}\! \left(x , y\right)+F_{37}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{38}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{19}\! \left(x \right) F_{27}\! \left(x \right)\\ F_{39}\! \left(x , y\right) &= F_{40}\! \left(x , y\right)\\ F_{40}\! \left(x , y\right) &= F_{27}\! \left(x \right) F_{30}\! \left(x , y\right) F_{36}\! \left(x , y\right)\\ F_{41}\! \left(x , y\right) &= F_{20}\! \left(x , y\right) F_{21}\! \left(x , y\right)\\ F_{42}\! \left(x \right) &= F_{34}\! \left(x , 1\right)\\ \end{align*}\)

This specification was found using the strategy pack "Point And Col Placements Tracked Fusion" and has 56 rules.

Found on July 23, 2021.

Finding the specification took 87 seconds.

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\(\begin{align*} F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{1}\! \left(x \right) &= 1\\ F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{4}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{4}\! \left(x \right) &= x\\ F_{5}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{7}\! \left(x , 1\right)\\ F_{7}\! \left(x , y\right) &= F_{11}\! \left(x , y\right)+F_{8}\! \left(x \right)+F_{9}\! \left(x , y\right)\\ F_{8}\! \left(x \right) &= 0\\ F_{9}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{4}\! \left(x \right)\\ F_{10}\! \left(x , y\right) &= -\frac{y \left(F_{7}\! \left(x , 1\right)-F_{7}\! \left(x , y\right)\right)}{-1+y}\\ F_{11}\! \left(x , y\right) &= F_{12}\! \left(x , y\right) F_{13}\! \left(x , y\right)\\ F_{12}\! \left(x , y\right) &= y x\\ F_{13}\! \left(x , y\right) &= F_{14}\! \left(x , y\right)+F_{55}\! \left(x , y\right)\\ F_{14}\! \left(x , y\right) &= F_{0}\! \left(x \right)+F_{15}\! \left(x , y\right)\\ F_{15}\! \left(x , y\right) &= F_{16}\! \left(x , y\right)\\ F_{16}\! \left(x , y\right) &= F_{12}\! \left(x , y\right) F_{17}\! \left(x , y\right)\\ F_{17}\! \left(x , y\right) &= F_{14}\! \left(x , y\right)+F_{18}\! \left(x , y\right)\\ F_{18}\! \left(x , y\right) &= F_{19}\! \left(x , y\right)\\ F_{19}\! \left(x , y\right) &= F_{20}\! \left(x , y\right) F_{4}\! \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_{22}\! \left(x , y\right)+F_{47}\! \left(x , y\right)\\ F_{22}\! \left(x , y\right) &= F_{23}\! \left(x \right) F_{44}\! \left(x , y\right)\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x , 1\right)\\ F_{25}\! \left(x , y\right) &= F_{24}\! \left(x , y\right)+F_{34}\! \left(x , y\right)\\ F_{25}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{26}\! \left(x , y\right)+F_{28}\! \left(x , y\right)+F_{29}\! \left(x \right)\\ F_{26}\! \left(x , y\right) &= F_{27}\! \left(x , y\right) F_{4}\! \left(x \right)\\ F_{27}\! \left(x , y\right) &= -\frac{-y F_{25}\! \left(x , y\right)+F_{25}\! \left(x , 1\right)}{-1+y}\\ F_{28}\! \left(x , y\right) &= F_{12}\! \left(x , y\right) F_{25}\! \left(x , y\right)\\ F_{29}\! \left(x \right) &= F_{30}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{29}\! \left(x \right)+F_{31}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{34}\! \left(x , 1\right)\\ F_{34}\! \left(x , y\right) &= F_{35}\! \left(x , y\right)\\ F_{35}\! \left(x , y\right) &= F_{25}\! \left(x , y\right) F_{36}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{1}\! \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 , 1\right)\\ F_{39}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{40}\! \left(x , y\right)+F_{43}\! \left(x , y\right)\\ F_{40}\! \left(x , y\right) &= F_{4}\! \left(x \right) F_{41}\! \left(x , y\right)\\ F_{41}\! \left(x , y\right) &= F_{42}\! \left(x , 1, y\right)\\ F_{42}\! \left(x , y , z\right) &= \frac{y F_{39}\! \left(x , y\right)-z F_{39}\! \left(x , z\right)}{-z +y}\\ F_{43}\! \left(x , y\right) &= F_{12}\! \left(x , y\right) F_{39}\! \left(x , y\right)\\ F_{44}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{45}\! \left(x , y\right)\\ F_{45}\! \left(x , y\right) &= F_{46}\! \left(x , y\right)\\ F_{46}\! \left(x , y\right) &= F_{12}\! \left(x , y\right) F_{44}\! \left(x , y\right)\\ F_{47}\! \left(x , y\right) &= F_{48}\! \left(x , y\right)\\ F_{48}\! \left(x , y\right) &= F_{23}\! \left(x \right) F_{4}\! \left(x \right) F_{49}\! \left(x , y\right)\\ F_{50}\! \left(x , y\right) &= F_{24}\! \left(x , y\right) F_{4}\! \left(x \right) F_{49}\! \left(x , y\right)\\ F_{50}\! \left(x , y\right) &= F_{51}\! \left(x , y\right)\\ F_{52}\! \left(x , y\right) &= F_{51}\! \left(x , y\right)+F_{54}\! \left(x , y\right)\\ F_{53}\! \left(x , y\right) &= F_{12}\! \left(x , y\right) F_{52}\! \left(x , y\right)\\ F_{53}\! \left(x , y\right) &= F_{7}\! \left(x , y\right)\\ F_{54}\! \left(x , y\right) &= F_{24}\! \left(x , y\right) F_{44}\! \left(x , y\right)\\ F_{55}\! \left(x , y\right) &= F_{7}\! \left(x , y\right)\\ \end{align*}\)

This specification was found using the strategy pack "Pattern Point Placements Tracked Fusion" and has 87 rules.

Found on July 23, 2021.

Finding the specification took 207 seconds.

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

This specification was found using the strategy pack "Insertion Point Row And Col Placements Tracked Fusion Req Corrob" and has 143 rules.

Found on July 23, 2021.

Finding the specification took 207 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_{22}\! \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_{48}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{22}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{46}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x , 1\right)\\ F_{10}\! \left(x , y\right) &= F_{11}\! \left(x , y\right) F_{22}\! \left(x \right)\\ F_{11}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{12}\! \left(x , y\right)+F_{43}\! \left(x , y\right)+F_{45}\! \left(x , y\right)\\ F_{12}\! \left(x , y\right) &= F_{13}\! \left(x , y\right) F_{22}\! \left(x \right)\\ F_{13}\! \left(x , y\right) &= F_{14}\! \left(x , y\right)+F_{26}\! \left(x , y\right)\\ F_{14}\! \left(x , y\right) &= F_{15}\! \left(x , y\right)+F_{19}\! \left(x , y\right)\\ F_{15}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{16}\! \left(x , y\right)\\ F_{16}\! \left(x , y\right) &= F_{17}\! \left(x , y\right)\\ F_{17}\! \left(x , y\right) &= F_{15}\! \left(x , y\right) F_{18}\! \left(x , y\right)\\ F_{18}\! \left(x , y\right) &= y x\\ F_{19}\! \left(x , y\right) &= F_{20}\! \left(x \right)+F_{24}\! \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_{23}\! \left(x \right)\\ F_{22}\! \left(x \right) &= x\\ F_{23}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{20}\! \left(x \right)\\ F_{24}\! \left(x , y\right) &= F_{25}\! \left(x , y\right)\\ F_{25}\! \left(x , y\right) &= F_{18}\! \left(x , y\right) F_{19}\! \left(x , y\right)\\ F_{26}\! \left(x , y\right) &= F_{27}\! \left(x , y\right)+F_{33}\! \left(x , y\right)\\ F_{27}\! \left(x , y\right) &= F_{20}\! \left(x \right)+F_{28}\! \left(x , y\right)\\ F_{28}\! \left(x , y\right) &= F_{29}\! \left(x \right)+F_{30}\! \left(x , y\right)+F_{32}\! \left(x , y\right)\\ F_{29}\! \left(x \right) &= 0\\ F_{30}\! \left(x , y\right) &= F_{22}\! \left(x \right) F_{31}\! \left(x , y\right)\\ F_{31}\! \left(x , y\right) &= F_{16}\! \left(x , y\right)+F_{28}\! \left(x , y\right)\\ F_{32}\! \left(x , y\right) &= F_{18}\! \left(x , y\right) F_{27}\! \left(x , y\right)\\ F_{33}\! \left(x , y\right) &= F_{34}\! \left(x \right)+F_{39}\! \left(x , y\right)\\ F_{34}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{35}\! \left(x \right)+F_{37}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{22}\! \left(x \right) F_{36}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{34}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{22}\! \left(x \right) F_{38}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{34}\! \left(x \right)\\ F_{39}\! \left(x , y\right) &= 2 F_{29}\! \left(x \right)+F_{40}\! \left(x , y\right)+F_{42}\! \left(x , y\right)\\ F_{40}\! \left(x , y\right) &= F_{22}\! \left(x \right) F_{41}\! \left(x , y\right)\\ F_{41}\! \left(x , y\right) &= F_{24}\! \left(x , y\right)+F_{39}\! \left(x , y\right)\\ F_{42}\! \left(x , y\right) &= F_{18}\! \left(x , y\right) F_{33}\! \left(x , y\right)\\ F_{43}\! \left(x , y\right) &= F_{22}\! \left(x \right) F_{44}\! \left(x , y\right)\\ F_{44}\! \left(x , y\right) &= -\frac{-y F_{11}\! \left(x , y\right)+F_{11}\! \left(x , 1\right)}{-1+y}\\ F_{45}\! \left(x , y\right) &= F_{11}\! \left(x , y\right) F_{18}\! \left(x , y\right)\\ F_{46}\! \left(x \right) &= F_{22}\! \left(x \right) F_{47}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{38}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{22}\! \left(x \right) F_{50}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{129}\! \left(x \right)+F_{51}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{23}\! \left(x \right) F_{52}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{53}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{54}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{55}\! \left(x , 1\right)\\ F_{55}\! \left(x , y\right) &= F_{56}\! \left(x , y\right)\\ F_{56}\! \left(x , y\right) &= F_{18}\! \left(x , y\right) F_{57}\! \left(x , y\right)\\ F_{57}\! \left(x , y\right) &= F_{58}\! \left(x , y\right)+F_{59}\! \left(x , y\right)\\ F_{58}\! \left(x , y\right) &= F_{2}\! \left(x \right)+F_{55}\! \left(x , y\right)\\ F_{59}\! \left(x , y\right) &= F_{60}\! \left(x , y\right)+F_{61}\! \left(x , y\right)\\ F_{60}\! \left(x , y\right) &= F_{15}\! \left(x , y\right) F_{6}\! \left(x \right)\\ F_{61}\! \left(x , y\right) &= F_{48}\! \left(x \right)+F_{62}\! \left(x , y\right)\\ F_{62}\! \left(x , y\right) &= F_{63}\! \left(x , y\right)\\ F_{63}\! \left(x , y\right) &= F_{22}\! \left(x \right) F_{64}\! \left(x , y\right)\\ F_{64}\! \left(x , y\right) &= F_{65}\! \left(x , y\right)+F_{68}\! \left(x , y\right)\\ F_{65}\! \left(x , y\right) &= F_{23}\! \left(x \right) F_{66}\! \left(x , y\right)\\ F_{66}\! \left(x , y\right) &= F_{24}\! \left(x , y\right)+F_{67}\! \left(x , y\right)\\ F_{67}\! \left(x , y\right) &= -\frac{y \left(F_{55}\! \left(x , 1\right)-F_{55}\! \left(x , y\right)\right)}{-1+y}\\ F_{68}\! \left(x , y\right) &= F_{69}\! \left(x , y\right)+F_{79}\! \left(x , y\right)\\ F_{69}\! \left(x , y\right) &= F_{70}\! \left(x , y\right)\\ F_{70}\! \left(x , y\right) &= F_{16}\! \left(x , y\right) F_{20}\! \left(x \right) F_{71}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{6}\! \left(x \right)+F_{72}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{73}\! \left(x , 1\right)\\ F_{73}\! \left(x , y\right) &= F_{29}\! \left(x \right)+F_{74}\! \left(x , y\right)+F_{77}\! \left(x , y\right)\\ F_{74}\! \left(x , y\right) &= -\frac{y \left(F_{75}\! \left(x , 1\right)-F_{75}\! \left(x , y\right)\right)}{-1+y}\\ F_{75}\! \left(x , y\right) &= F_{22}\! \left(x \right) F_{76}\! \left(x , y\right)\\ F_{76}\! \left(x , y\right) &= F_{16}\! \left(x , y\right)+F_{73}\! \left(x , y\right)\\ F_{77}\! \left(x , y\right) &= F_{18}\! \left(x , y\right) F_{78}\! \left(x , y\right)\\ F_{78}\! \left(x , y\right) &= F_{6}\! \left(x \right)+F_{73}\! \left(x , y\right)\\ F_{79}\! \left(x , y\right) &= -\frac{y \left(F_{80}\! \left(x , 1\right)-F_{80}\! \left(x , y\right)\right)}{-1+y}\\ F_{80}\! \left(x , y\right) &= F_{62}\! \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_{22}\! \left(x \right) F_{83}\! \left(x , y\right)\\ F_{83}\! \left(x , y\right) &= F_{84}\! \left(x , y\right)+F_{87}\! \left(x , y\right)\\ F_{84}\! \left(x , y\right) &= F_{72}\! \left(x \right) F_{85}\! \left(x , y\right)\\ F_{86}\! \left(x , y\right) &= F_{22}\! \left(x \right) F_{85}\! \left(x , y\right)\\ F_{86}\! \left(x , y\right) &= F_{55}\! \left(x , y\right)\\ F_{87}\! \left(x , y\right) &= F_{128}\! \left(x \right) F_{88}\! \left(x , y\right)\\ F_{88}\! \left(x , y\right) &= F_{89}\! \left(x , y\right)+F_{90}\! \left(x , y\right)\\ F_{89}\! \left(x , y\right) &= F_{16}\! \left(x , y\right) F_{5}\! \left(x \right)\\ F_{90}\! \left(x , y\right) &= F_{91}\! \left(x , y\right)+F_{99}\! \left(x , y\right)\\ F_{91}\! \left(x , y\right) &= F_{92}\! \left(x , y\right)\\ F_{92}\! \left(x , y\right) &= F_{6}\! \left(x \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_{18}\! \left(x , y\right) F_{95}\! \left(x , y\right)\\ F_{95}\! \left(x , y\right) &= F_{60}\! \left(x , y\right)+F_{96}\! \left(x , y\right)\\ F_{96}\! \left(x , y\right) &= F_{97}\! \left(x , y\right) F_{98}\! \left(x \right)\\ F_{97}\! \left(x , y\right) &= F_{6}\! \left(x \right)+F_{93}\! \left(x , y\right)\\ F_{98}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{100}\! \left(x , y\right) &= F_{107}\! \left(x \right)+F_{99}\! \left(x , y\right)\\ F_{101}\! \left(x , y\right) &= F_{100}\! \left(x , y\right)+F_{105}\! \left(x , y\right)\\ F_{102}\! \left(x , y\right) &= F_{101}\! \left(x , y\right)+F_{104}\! \left(x , y\right)\\ F_{103}\! \left(x , y\right) &= F_{102}\! \left(x , y\right) F_{18}\! \left(x , y\right)\\ F_{103}\! \left(x , y\right) &= F_{62}\! \left(x , y\right)\\ F_{104}\! \left(x , y\right) &= F_{15}\! \left(x , y\right) F_{48}\! \left(x \right)\\ F_{105}\! \left(x , y\right) &= F_{106}\! \left(x , y\right)\\ F_{106}\! \left(x , y\right) &= F_{6}\! \left(x \right) F_{97}\! \left(x , y\right)\\ F_{107}\! \left(x \right) &= -F_{126}\! \left(x \right)+F_{108}\! \left(x \right)\\ F_{108}\! \left(x \right) &= -F_{5}\! \left(x \right)+F_{109}\! \left(x \right)\\ F_{109}\! \left(x \right) &= \frac{F_{110}\! \left(x \right)}{F_{22}\! \left(x \right)}\\ F_{110}\! \left(x \right) &= -F_{111}\! \left(x \right)-F_{29}\! \left(x \right)+F_{48}\! \left(x \right)\\ F_{111}\! \left(x \right) &= F_{112}\! \left(x \right) F_{22}\! \left(x \right)\\ F_{112}\! \left(x \right) &= F_{113}\! \left(x \right)+F_{114}\! \left(x \right)\\ F_{113}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{48}\! \left(x \right)\\ F_{114}\! \left(x \right) &= F_{115}\! \left(x \right)+F_{117}\! \left(x \right)\\ F_{115}\! \left(x \right) &= F_{116}\! \left(x \right)\\ F_{116}\! \left(x \right) &= F_{2}\! \left(x \right) F_{20}\! \left(x \right)\\ F_{117}\! \left(x \right) &= F_{118}\! \left(x , 1\right)\\ F_{118}\! \left(x , y\right) &= 2 F_{29}\! \left(x \right)+F_{119}\! \left(x , y\right)+F_{121}\! \left(x , y\right)\\ F_{119}\! \left(x , y\right) &= F_{120}\! \left(x , y\right) F_{18}\! \left(x , y\right)\\ F_{120}\! \left(x , y\right) &= F_{118}\! \left(x , y\right)+F_{48}\! \left(x \right)\\ F_{121}\! \left(x , y\right) &= -\frac{y \left(F_{122}\! \left(x , 1\right)-F_{122}\! \left(x , y\right)\right)}{-1+y}\\ F_{122}\! \left(x , y\right) &= F_{123}\! \left(x , y\right) F_{22}\! \left(x \right)\\ F_{123}\! \left(x , y\right) &= F_{118}\! \left(x , y\right)+F_{124}\! \left(x , y\right)\\ F_{124}\! \left(x , y\right) &= F_{125}\! \left(x , y\right)\\ F_{125}\! \left(x , y\right) &= F_{16}\! \left(x , y\right) F_{2}\! \left(x \right)\\ F_{126}\! \left(x \right) &= F_{127}\! \left(x \right)\\ F_{127}\! \left(x \right) &= F_{6} \left(x \right)^{2}\\ F_{128}\! \left(x \right) &= F_{76}\! \left(x , 1\right)\\ F_{129}\! \left(x \right) &= F_{130}\! \left(x \right)+F_{131}\! \left(x \right)\\ F_{130}\! \left(x \right) &= F_{20}\! \left(x \right) F_{71}\! \left(x \right)\\ F_{131}\! \left(x \right) &= F_{132}\! \left(x \right)+F_{134}\! \left(x \right)\\ F_{132}\! \left(x \right) &= F_{133}\! \left(x \right)+F_{48}\! \left(x \right)\\ F_{133}\! \left(x \right) &= F_{62}\! \left(x , 1\right)\\ F_{134}\! \left(x \right) &= F_{135}\! \left(x \right)+F_{142}\! \left(x \right)\\ F_{135}\! \left(x \right) &= F_{136}\! \left(x , 1\right)\\ F_{136}\! \left(x , y\right) &= F_{137}\! \left(x , y\right)+F_{140}\! \left(x , y\right)+F_{29}\! \left(x \right)\\ F_{137}\! \left(x , y\right) &= -\frac{y \left(F_{138}\! \left(x , 1\right)-F_{138}\! \left(x , y\right)\right)}{-1+y}\\ F_{138}\! \left(x , y\right) &= F_{139}\! \left(x , y\right) F_{22}\! \left(x \right)\\ F_{139}\! \left(x , y\right) &= F_{125}\! \left(x , y\right)+F_{136}\! \left(x , y\right)\\ F_{140}\! \left(x , y\right) &= F_{141}\! \left(x , y\right) F_{18}\! \left(x , y\right)\\ F_{141}\! \left(x , y\right) &= F_{136}\! \left(x , y\right)+F_{48}\! \left(x \right)\\ F_{142}\! \left(x \right) &= F_{81}\! \left(x , 1\right)\\ \end{align*}\)

This specification was found using the strategy pack "All The Strategies 2 Tracked Fusion" and has 84 rules.

Found on July 23, 2021.

Finding the specification took 187 seconds.

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