Av(12435, 12453, 12543, 21435, 21453, 21543)
Generating Function
\(\displaystyle \frac{\left(-7 x^{2}+x \right) \sqrt{x^{2}-6 x +1}-3 x^{3}-4 x^{2}+13 x -2}{20 x^{3}-26 x^{2}+16 x -2}\)
Counting Sequence
1, 1, 2, 6, 24, 114, 596, 3290, 18760, 109186, 644188, 3836682, 23005904, 138641666, 838667012, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(10 x^{3}-13 x^{2}+8 x -1\right) F \left(x
\right)^{2}+\left(3 x^{3}+4 x^{2}-13 x +2\right) F \! \left(x \right)-x^{3}+7 x^{2}+5 x -1 = 0\)
Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 114\)
\(\displaystyle a{\left(n + 6 \right)} = - \frac{70 n a{\left(n \right)}}{n + 5} + \frac{\left(21 n + 82\right) a{\left(n + 5 \right)}}{n + 5} - \frac{2 \left(80 n + 233\right) a{\left(n + 4 \right)}}{n + 5} - \frac{5 \left(149 n + 253\right) a{\left(n + 2 \right)}}{n + 5} + \frac{2 \left(265 n + 558\right) a{\left(n + 3 \right)}}{n + 5} + \frac{\left(521 n + 650\right) a{\left(n + 1 \right)}}{n + 5}, \quad n \geq 6\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 114\)
\(\displaystyle a{\left(n + 6 \right)} = - \frac{70 n a{\left(n \right)}}{n + 5} + \frac{\left(21 n + 82\right) a{\left(n + 5 \right)}}{n + 5} - \frac{2 \left(80 n + 233\right) a{\left(n + 4 \right)}}{n + 5} - \frac{5 \left(149 n + 253\right) a{\left(n + 2 \right)}}{n + 5} + \frac{2 \left(265 n + 558\right) a{\left(n + 3 \right)}}{n + 5} + \frac{\left(521 n + 650\right) a{\left(n + 1 \right)}}{n + 5}, \quad n \geq 6\)
This specification was found using the strategy pack "Point Placements Req Corrob" and has 62 rules.
Finding the specification took 2155 seconds.
Copy 62 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_{17}\! \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_{7}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{6}\! \left(x \right) x +F_{6} \left(x \right)^{2}+x\\
F_{7}\! \left(x \right) &= F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{17}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{13}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right) F_{2}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{12}\! \left(x \right) x +F_{12} \left(x \right)^{2}-2 F_{12}\! \left(x \right)+2\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right) F_{17}\! \left(x \right) F_{18}\! \left(x \right)\\
F_{15}\! \left(x \right) &= \frac{F_{16}\! \left(x \right)}{F_{17}\! \left(x \right)}\\
F_{16}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{17}\! \left(x \right) &= x\\
F_{18}\! \left(x \right) &= \frac{F_{19}\! \left(x \right)}{F_{2}\! \left(x \right)}\\
F_{19}\! \left(x \right) &= -F_{22}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= \frac{F_{21}\! \left(x \right)}{F_{17}\! \left(x \right)}\\
F_{21}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{17}\! \left(x \right) F_{24}\! \left(x \right) F_{26}\! \left(x \right)\\
F_{24}\! \left(x \right) &= \frac{F_{25}\! \left(x \right)}{F_{17}\! \left(x \right)}\\
F_{25}\! \left(x \right) &= F_{5}\! \left(x \right)\\
F_{26}\! \left(x \right) &= \frac{F_{27}\! \left(x \right)}{F_{17}\! \left(x \right) F_{24}\! \left(x \right)}\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= -F_{31}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= \frac{F_{30}\! \left(x \right)}{F_{17}\! \left(x \right)}\\
F_{30}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{4}\! \left(x \right) F_{6}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{17}\! \left(x \right) F_{24}\! \left(x \right) F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{17}\! \left(x \right) F_{26}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{17}\! \left(x \right) F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{17}\! \left(x \right) F_{24}\! \left(x \right) F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= \frac{F_{43}\! \left(x \right)}{F_{17}\! \left(x \right)}\\
F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= \frac{F_{45}\! \left(x \right)}{F_{17}\! \left(x \right)}\\
F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{17}\! \left(x \right) F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{17}\! \left(x \right) F_{26}\! \left(x \right) F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= \frac{F_{50}\! \left(x \right)}{F_{0}\! \left(x \right)}\\
F_{50}\! \left(x \right) &= -F_{58}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= \frac{F_{52}\! \left(x \right)}{F_{17}\! \left(x \right)}\\
F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{54}\! \left(x \right) &= -F_{2}\! \left(x \right)+F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= -F_{12}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= \frac{F_{57}\! \left(x \right)}{F_{17}\! \left(x \right)}\\
F_{57}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{58}\! \left(x \right) &= -F_{61}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= \frac{F_{60}\! \left(x \right)}{F_{17}\! \left(x \right)}\\
F_{60}\! \left(x \right) &= F_{54}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{2}\! \left(x \right) F_{49}\! \left(x \right)\\
\end{align*}\)
This specification was found using the strategy pack "Point Placements Tracked Fusion" and has 21 rules.
Finding the specification took 12 seconds.
Copy 21 equations to clipboard:
\(\begin{align*}
F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{1}\! \left(x \right) &= 1\\
F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\
F_{3}\! \left(x \right) &= F_{18}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{18}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{0}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x , 1\right)\\
F_{11}\! \left(x , y\right) &= F_{10}\! \left(x , y\right) F_{13}\! \left(x , y\right)\\
F_{11}\! \left(x , y\right) &= F_{12}\! \left(x , y\right)\\
F_{12}\! \left(x , y\right) &= x F_{12}\! \left(x , y\right) y +y x +F_{12}\! \left(x , y\right)^{2}\\
F_{13}\! \left(x , y\right) &= y x\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right) F_{18}\! \left(x \right) F_{19}\! \left(x \right)\\
F_{16}\! \left(x \right) &= \frac{F_{17}\! \left(x \right)}{F_{18}\! \left(x \right)}\\
F_{17}\! \left(x \right) &= F_{5}\! \left(x \right)\\
F_{18}\! \left(x \right) &= x\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x , 1\right)\\
F_{20}\! \left(x , y\right) &= -\frac{-y F_{10}\! \left(x , y\right)+F_{10}\! \left(x , 1\right)}{-1+y}\\
\end{align*}\)
This specification was found using the strategy pack "Point Placements Req Corrob" and has 78 rules.
Finding the specification took 979 seconds.
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Copy 78 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_{20}\! \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_{20}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= \frac{F_{8}\! \left(x \right)}{F_{20}\! \left(x \right)}\\
F_{8}\! \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_{20}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{15}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{0}\! \left(x \right) F_{14}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{14}\! \left(x \right) x +F_{14} \left(x \right)^{2}-2 F_{14}\! \left(x \right)+2\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right) F_{20}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right) F_{21}\! \left(x \right)\\
F_{20}\! \left(x \right) &= x\\
F_{21}\! \left(x \right) &= \frac{F_{22}\! \left(x \right)}{F_{20}\! \left(x \right) F_{42}\! \left(x \right)}\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= -F_{28}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= \frac{F_{25}\! \left(x \right)}{F_{20}\! \left(x \right)}\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= -F_{27}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{27}\! \left(x \right) x +F_{27} \left(x \right)^{2}+x\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{2}\! \left(x \right) F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= \frac{F_{31}\! \left(x \right)}{F_{0}\! \left(x \right)}\\
F_{31}\! \left(x \right) &= -F_{34}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= \frac{F_{33}\! \left(x \right)}{F_{20}\! \left(x \right)}\\
F_{33}\! \left(x \right) &= F_{9}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{20}\! \left(x \right) F_{21}\! \left(x \right) F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= \frac{F_{38}\! \left(x \right)}{F_{20}\! \left(x \right)}\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= -F_{2}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= -F_{41}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{41}\! \left(x \right) &= x^{2} F_{41} \left(x \right)^{2}+2 x^{2} F_{41}\! \left(x \right)+4 x F_{41} \left(x \right)^{2}+x^{2}-13 x F_{41}\! \left(x \right)-F_{41} \left(x \right)^{2}+8 x +4 F_{41}\! \left(x \right)-2\\
F_{42}\! \left(x \right) &= \frac{F_{43}\! \left(x \right)}{F_{0}\! \left(x \right)}\\
F_{43}\! \left(x \right) &= -F_{52}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= \frac{F_{45}\! \left(x \right)}{F_{20}\! \left(x \right)}\\
F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{48}\! \left(x \right)\\
F_{47}\! \left(x \right) &= x^{2} F_{47} \left(x \right)^{2}+4 x^{2} F_{47}\! \left(x \right)+4 x F_{47} \left(x \right)^{2}+4 x^{2}-5 x F_{47}\! \left(x \right)-F_{47} \left(x \right)^{2}-x +2 F_{47}\! \left(x \right)\\
F_{48}\! \left(x \right) &= -F_{2}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= -F_{41}\! \left(x \right)+F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= \frac{F_{51}\! \left(x \right)}{F_{20}\! \left(x \right)}\\
F_{51}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{52}\! \left(x \right) &= -F_{55}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= \frac{F_{54}\! \left(x \right)}{F_{20}\! \left(x \right)}\\
F_{54}\! \left(x \right) &= F_{48}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{2}\! \left(x \right) F_{42}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{20}\! \left(x \right) F_{30}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{20}\! \left(x \right) F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{20}\! \left(x \right) F_{63}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{63}\! \left(x \right) &= \frac{F_{64}\! \left(x \right)}{F_{20}\! \left(x \right)}\\
F_{64}\! \left(x \right) &= F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{20}\! \left(x \right) F_{21}\! \left(x \right) F_{66}\! \left(x \right)\\
F_{66}\! \left(x \right) &= \frac{F_{67}\! \left(x \right)}{F_{0}\! \left(x \right)}\\
F_{67}\! \left(x \right) &= -F_{73}\! \left(x \right)+F_{68}\! \left(x \right)\\
F_{68}\! \left(x \right) &= \frac{F_{69}\! \left(x \right)}{F_{20}\! \left(x \right)}\\
F_{69}\! \left(x \right) &= F_{70}\! \left(x \right)\\
F_{70}\! \left(x \right) &= -F_{0}\! \left(x \right)+F_{71}\! \left(x \right)\\
F_{71}\! \left(x \right) &= \frac{F_{72}\! \left(x \right)}{F_{20}\! \left(x \right)}\\
F_{72}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{73}\! \left(x \right) &= -F_{77}\! \left(x \right)+F_{74}\! \left(x \right)\\
F_{74}\! \left(x \right) &= \frac{F_{75}\! \left(x \right)}{F_{20}\! \left(x \right)}\\
F_{75}\! \left(x \right) &= F_{76}\! \left(x \right)\\
F_{76}\! \left(x \right) &= -F_{27}\! \left(x \right)+F_{70}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{2}\! \left(x \right) F_{66}\! \left(x \right)\\
\end{align*}\)