Av(1342, 2413, 3124)
Generating Function
\(\displaystyle -\frac{\left(2 x -1\right) \left(x^{2}-3 x +1\right)^{2}}{\left(x -1\right) \left(x^{5}-8 x^{4}+24 x^{3}-22 x^{2}+8 x -1\right)}\)
Counting Sequence
1, 1, 2, 6, 21, 76, 274, 977, 3449, 12086, 42141, 146469, 508098, 1760610, 6096937, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x -1\right) \left(x^{5}-8 x^{4}+24 x^{3}-22 x^{2}+8 x -1\right) F \! \left(x \right)+\left(2 x -1\right) \left(x^{2}-3 x +1\right)^{2} = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 76\)
\(\displaystyle a \! \left(n +5\right) = a \! \left(n \right)-8 a \! \left(n +1\right)+24 a \! \left(n +2\right)-22 a \! \left(n +3\right)+8 a \! \left(n +4\right)-1, \quad n \geq 6\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 76\)
\(\displaystyle a \! \left(n +5\right) = a \! \left(n \right)-8 a \! \left(n +1\right)+24 a \! \left(n +2\right)-22 a \! \left(n +3\right)+8 a \! \left(n +4\right)-1, \quad n \geq 6\)
Explicit Closed Form
\(\displaystyle \frac{1}{2}+\frac{5341 \mathit{RootOf} \left(Z^{5}-8 Z^{4}+24 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =1\right)^{-n +4}}{9514}+\frac{5341 \mathit{RootOf} \left(Z^{5}-8 Z^{4}+24 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =2\right)^{-n +4}}{9514}+\frac{5341 \mathit{RootOf} \left(Z^{5}-8 Z^{4}+24 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =3\right)^{-n +4}}{9514}+\frac{5341 \mathit{RootOf} \left(Z^{5}-8 Z^{4}+24 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =4\right)^{-n +4}}{9514}+\frac{5341 \mathit{RootOf} \left(Z^{5}-8 Z^{4}+24 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =5\right)^{-n +4}}{9514}-\frac{44137 \mathit{RootOf} \left(Z^{5}-8 Z^{4}+24 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =1\right)^{-n +3}}{9514}-\frac{44137 \mathit{RootOf} \left(Z^{5}-8 Z^{4}+24 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =2\right)^{-n +3}}{9514}-\frac{44137 \mathit{RootOf} \left(Z^{5}-8 Z^{4}+24 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =3\right)^{-n +3}}{9514}-\frac{44137 \mathit{RootOf} \left(Z^{5}-8 Z^{4}+24 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =4\right)^{-n +3}}{9514}-\frac{44137 \mathit{RootOf} \left(Z^{5}-8 Z^{4}+24 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =5\right)^{-n +3}}{9514}+\frac{138343 \mathit{RootOf} \left(Z^{5}-8 Z^{4}+24 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =1\right)^{-n +2}}{9514}+\frac{138343 \mathit{RootOf} \left(Z^{5}-8 Z^{4}+24 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =2\right)^{-n +2}}{9514}+\frac{138343 \mathit{RootOf} \left(Z^{5}-8 Z^{4}+24 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =3\right)^{-n +2}}{9514}+\frac{138343 \mathit{RootOf} \left(Z^{5}-8 Z^{4}+24 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =4\right)^{-n +2}}{9514}+\frac{138343 \mathit{RootOf} \left(Z^{5}-8 Z^{4}+24 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =5\right)^{-n +2}}{9514}-\frac{143489 \mathit{RootOf} \left(Z^{5}-8 Z^{4}+24 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =1\right)^{-n +1}}{9514}-\frac{143489 \mathit{RootOf} \left(Z^{5}-8 Z^{4}+24 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =2\right)^{-n +1}}{9514}-\frac{143489 \mathit{RootOf} \left(Z^{5}-8 Z^{4}+24 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =3\right)^{-n +1}}{9514}-\frac{143489 \mathit{RootOf} \left(Z^{5}-8 Z^{4}+24 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =4\right)^{-n +1}}{9514}-\frac{143489 \mathit{RootOf} \left(Z^{5}-8 Z^{4}+24 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =5\right)^{-n +1}}{9514}-\frac{3276 \mathit{RootOf} \left(Z^{5}-8 Z^{4}+24 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =1\right)^{-n -1}}{4757}-\frac{3276 \mathit{RootOf} \left(Z^{5}-8 Z^{4}+24 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =2\right)^{-n -1}}{4757}-\frac{3276 \mathit{RootOf} \left(Z^{5}-8 Z^{4}+24 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =3\right)^{-n -1}}{4757}-\frac{3276 \mathit{RootOf} \left(Z^{5}-8 Z^{4}+24 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =4\right)^{-n -1}}{4757}-\frac{3276 \mathit{RootOf} \left(Z^{5}-8 Z^{4}+24 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =5\right)^{-n -1}}{4757}+\frac{55251 \mathit{RootOf} \left(Z^{5}-8 Z^{4}+24 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =1\right)^{-n}}{9514}+\frac{55251 \mathit{RootOf} \left(Z^{5}-8 Z^{4}+24 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =2\right)^{-n}}{9514}+\frac{55251 \mathit{RootOf} \left(Z^{5}-8 Z^{4}+24 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =3\right)^{-n}}{9514}+\frac{55251 \mathit{RootOf} \left(Z^{5}-8 Z^{4}+24 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =4\right)^{-n}}{9514}+\frac{55251 \mathit{RootOf} \left(Z^{5}-8 Z^{4}+24 Z^{3}-22 Z^{2}+8 Z -1, \mathit{index} =5\right)^{-n}}{9514}\)
This specification was found using the strategy pack "Insertion Point Placements" and has 74 rules.
Found on July 23, 2021.Finding the specification took 10 seconds.
Copy 74 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_{16}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\
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_{10}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \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 \right)+F_{15}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{6}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{70}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{68}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \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_{31}\! \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_{27}\! \left(x \right) &= F_{25}\! \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_{29}\! \left(x \right) &= F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{28}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{34}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{33}\! \left(x \right) &= 0\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{41}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{39}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{44}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{47}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)+F_{66}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{28}\! \left(x \right) F_{6}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{22}\! \left(x \right) F_{6}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{55}\! \left(x \right) F_{6}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{57}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{55}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{22}\! \left(x \right) F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{63}\! \left(x \right)+F_{65}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{64}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{62}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{61}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{67}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{22}\! \left(x \right) F_{28}\! \left(x \right) F_{49}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{69}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{22}\! \left(x \right) F_{26}\! \left(x \right) F_{5}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{2}\! \left(x \right) F_{71}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{72}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{73}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{25}\! \left(x \right) F_{5}\! \left(x \right)\\
\end{align*}\)