Av(14325, 14352, 15324, 15342, 24315, 24351, 25314, 25341, 41325, 41352, 42315, 42351, 51324, 51342, 52314, 52341, 132546, 132564, 142536, 142563, 312546, 312564, 365214, 365241, 412536, 412563, 465213, 465231, 635214, 635241, 645213, 645231)
View Raw Data
Generating Function
\(\displaystyle \frac{\left(3 x -1\right) \left(2 x^{2}-4 x +1\right)}{\left(4 x -1\right) \left(2 x -1\right)^{2}}\)
Counting Sequence
1, 1, 2, 6, 24, 104, 448, 1888, 7808, 31872, 129024, 519680, 2086912, 8366080, 33505280, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(4 x -1\right) \left(2 x -1\right)^{2} F \! \left(x \right)-\left(3 x -1\right) \left(2 x^{2}-4 x +1\right) = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(n +3\right) = 16 a \! \left(n \right)-20 a \! \left(n +1\right)+8 a \! \left(n +2\right), \quad n \geq 4\)
Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0 \\ \frac{4^{n}}{8}-\frac{2^{n} n}{4}+2^{-1+n} & \text{otherwise} \end{array}\right.\)

This specification was found using the strategy pack "Col Placements Tracked Fusion Req Corrob" and has 110 rules.

Found on July 23, 2021.

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

This specification was found using the strategy pack "Col Placements Tracked Fusion Req Corrob" and has 28 rules.

Found on July 23, 2021.

Finding the specification took 6 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_{11}\! \left(x \right) F_{3}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{27}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{11}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{24}\! \left(x \right)+F_{26}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{11}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{8}\! \left(x , 1\right)\\ F_{8}\! \left(x , y_{0}\right) &= -\frac{-y_{0} F_{9}\! \left(x , y_{0}\right)+F_{9}\! \left(x , 1\right)}{-1+y_{0}}\\ F_{9}\! \left(x , y_{0}\right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x , y_{0}\right)+F_{12}\! \left(x , y_{0}\right)+F_{22}\! \left(x , y_{0}\right)\\ F_{10}\! \left(x , y_{0}\right) &= F_{11}\! \left(x \right) F_{8}\! \left(x , y_{0}\right)\\ F_{11}\! \left(x \right) &= x\\ F_{12}\! \left(x , y_{0}\right) &= F_{13}\! \left(x , y_{0}\right)\\ F_{13}\! \left(x , y_{0}\right) &= F_{14}\! \left(x , y_{0}\right) F_{20}\! \left(x , y_{0}\right)\\ F_{14}\! \left(x , y_{0}\right) &= \frac{y_{0} F_{15}\! \left(x , y_{0}, 1\right)-F_{15}\! \left(x , y_{0}, \frac{1}{y_{0}}\right)}{-1+y_{0}}\\ F_{15}\! \left(x , y_{0}, y_{1}\right) &= \frac{y_{0} F_{16}\! \left(x , 1, y_{0}, y_{1}\right)-F_{16}\! \left(x , \frac{1}{y_{0}}, y_{0}, y_{1}\right)}{-1+y_{0}}\\ F_{16}\! \left(x , y_{0}, y_{1}, y_{2}\right) &= F_{17}\! \left(x , y_{0} y_{1}\right)+F_{18}\! \left(x , y_{1} y_{2}\right)\\ F_{17}\! \left(x , y_{0}\right) &= F_{1}\! \left(x \right)+F_{18}\! \left(x , y_{0}\right)\\ F_{18}\! \left(x , y_{0}\right) &= F_{19}\! \left(x , y_{0}\right)\\ F_{19}\! \left(x , y_{0}\right) &= F_{20}\! \left(x , y_{0}\right) F_{21}\! \left(x , y_{0}\right)\\ F_{20}\! \left(x , y_{0}\right) &= y_{0} x\\ F_{21}\! \left(x , y_{0}\right) &= 2 F_{19}\! \left(x , y_{0}\right)+F_{1}\! \left(x \right)\\ F_{22}\! \left(x , y_{0}\right) &= F_{11}\! \left(x \right) F_{23}\! \left(x , y_{0}\right)\\ F_{23}\! \left(x , y_{0}\right) &= -\frac{-y_{0} F_{9}\! \left(x , y_{0}\right)+F_{9}\! \left(x , 1\right)}{-1+y_{0}}\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{13}\! \left(x , 1\right)\\ F_{26}\! \left(x \right) &= F_{11}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{11}\! \left(x \right) F_{5}\! \left(x \right)\\ \end{align*}\)

This specification was found using the strategy pack "Col Placements Tracked Fusion Req Corrob" and has 29 rules.

Found on July 23, 2021.

Finding the specification took 6 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_{11}\! \left(x \right) F_{3}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{28}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{11}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{25}\! \left(x \right)+F_{27}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{11}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{8}\! \left(x , 1\right)\\ F_{8}\! \left(x , y\right) &= -\frac{-y F_{9}\! \left(x , y\right)+F_{9}\! \left(x , 1\right)}{-1+y}\\ F_{9}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x , y\right)+F_{12}\! \left(x , y\right)+F_{23}\! \left(x , y\right)\\ F_{10}\! \left(x , y\right) &= F_{11}\! \left(x \right) F_{8}\! \left(x , y\right)\\ F_{11}\! \left(x \right) &= x\\ 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_{21}\! \left(x , y\right)\\ F_{14}\! \left(x , y\right) &= \frac{y F_{15}\! \left(x , y , 1\right)-F_{15}\! \left(x , y , \frac{1}{y}\right)}{-1+y}\\ F_{15}\! \left(x , y , z\right) &= F_{16}\! \left(x , y , y z \right)\\ F_{16}\! \left(x , y , z\right) &= -\frac{-y F_{17}\! \left(x , y , z\right)+F_{17}\! \left(x , 1, z\right)}{-1+y}\\ F_{17}\! \left(x , y , z\right) &= F_{18}\! \left(x , y\right)+F_{19}\! \left(x , z\right)\\ F_{18}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{19}\! \left(x , y\right)\\ F_{19}\! \left(x , y\right) &= F_{20}\! \left(x , y\right)\\ F_{20}\! \left(x , y\right) &= F_{21}\! \left(x , y\right) F_{22}\! \left(x , y\right)\\ F_{21}\! \left(x , y\right) &= y x\\ F_{22}\! \left(x , y\right) &= 2 F_{20}\! \left(x , y\right)+F_{1}\! \left(x \right)\\ F_{23}\! \left(x , y\right) &= F_{11}\! \left(x \right) F_{24}\! \left(x , y\right)\\ F_{24}\! \left(x , y\right) &= -\frac{-y F_{9}\! \left(x , y\right)+F_{9}\! \left(x , 1\right)}{-1+y}\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{13}\! \left(x , 1\right)\\ F_{27}\! \left(x \right) &= F_{11}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{11}\! \left(x \right) F_{5}\! \left(x \right)\\ \end{align*}\)

This specification was found using the strategy pack "Col Placements Tracked Fusion Req Corrob" and has 110 rules.

Found on July 23, 2021.

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

This specification was found using the strategy pack "Col Placements Tracked Fusion Req Corrob" and has 29 rules.

Found on July 23, 2021.

Finding the specification took 6 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_{11}\! \left(x \right) F_{3}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{28}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{11}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{25}\! \left(x \right)+F_{27}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{11}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{8}\! \left(x , 1\right)\\ F_{8}\! \left(x , y\right) &= -\frac{-y F_{9}\! \left(x , y\right)+F_{9}\! \left(x , 1\right)}{-1+y}\\ F_{9}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x , y\right)+F_{12}\! \left(x , y\right)+F_{23}\! \left(x , y\right)\\ F_{10}\! \left(x , y\right) &= F_{11}\! \left(x \right) F_{8}\! \left(x , y\right)\\ F_{11}\! \left(x \right) &= x\\ 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_{21}\! \left(x , y\right)\\ F_{14}\! \left(x , y\right) &= \frac{y F_{15}\! \left(x , y , 1\right)-F_{15}\! \left(x , y , \frac{1}{y}\right)}{-1+y}\\ F_{15}\! \left(x , y , z\right) &= F_{16}\! \left(x , y , y z \right)\\ F_{16}\! \left(x , y , z\right) &= -\frac{-y F_{17}\! \left(x , y , z\right)+F_{17}\! \left(x , 1, z\right)}{-1+y}\\ F_{17}\! \left(x , y , z\right) &= F_{18}\! \left(x , y\right)+F_{19}\! \left(x , z\right)\\ F_{18}\! \left(x , y\right) &= F_{1}\! \left(x \right)+F_{19}\! \left(x , y\right)\\ F_{19}\! \left(x , y\right) &= F_{20}\! \left(x , y\right)\\ F_{20}\! \left(x , y\right) &= F_{21}\! \left(x , y\right) F_{22}\! \left(x , y\right)\\ F_{21}\! \left(x , y\right) &= y x\\ F_{22}\! \left(x , y\right) &= 2 F_{20}\! \left(x , y\right)+F_{1}\! \left(x \right)\\ F_{23}\! \left(x , y\right) &= F_{11}\! \left(x \right) F_{24}\! \left(x , y\right)\\ F_{24}\! \left(x , y\right) &= -\frac{-y F_{9}\! \left(x , y\right)+F_{9}\! \left(x , 1\right)}{-1+y}\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{13}\! \left(x , 1\right)\\ F_{27}\! \left(x \right) &= F_{11}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{11}\! \left(x \right) F_{5}\! \left(x \right)\\ \end{align*}\)