Av(1342, 3142, 3214)
Generating Function
\(\displaystyle -\frac{\left(x -1\right) \left(2 x -1\right) \left(x^{2}-3 x +1\right)}{x^{5}-7 x^{4}+18 x^{3}-17 x^{2}+7 x -1}\)
Counting Sequence
1, 1, 2, 6, 21, 75, 263, 904, 3066, 10324, 34652, 116179, 389443, 1305592, 4377595, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{5}-7 x^{4}+18 x^{3}-17 x^{2}+7 x -1\right) F \! \left(x \right)+\left(x -1\right) \left(2 x -1\right) \left(x^{2}-3 x +1\right) = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(n +5\right) = a \! \left(n \right)-7 a \! \left(n +1\right)+18 a \! \left(n +2\right)-17 a \! \left(n +3\right)+7 a \! \left(n +4\right), \quad n \geq 5\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(n +5\right) = a \! \left(n \right)-7 a \! \left(n +1\right)+18 a \! \left(n +2\right)-17 a \! \left(n +3\right)+7 a \! \left(n +4\right), \quad n \geq 5\)
Explicit Closed Form
\(\displaystyle -\frac{640 \mathit{RootOf} \left(Z^{5}-7 Z^{4}+18 Z^{3}-17 Z^{2}+7 Z -1, \mathit{index} =1\right)^{-n +3}}{3857}-\frac{640 \mathit{RootOf} \left(Z^{5}-7 Z^{4}+18 Z^{3}-17 Z^{2}+7 Z -1, \mathit{index} =2\right)^{-n +3}}{3857}-\frac{640 \mathit{RootOf} \left(Z^{5}-7 Z^{4}+18 Z^{3}-17 Z^{2}+7 Z -1, \mathit{index} =3\right)^{-n +3}}{3857}-\frac{640 \mathit{RootOf} \left(Z^{5}-7 Z^{4}+18 Z^{3}-17 Z^{2}+7 Z -1, \mathit{index} =4\right)^{-n +3}}{3857}-\frac{640 \mathit{RootOf} \left(Z^{5}-7 Z^{4}+18 Z^{3}-17 Z^{2}+7 Z -1, \mathit{index} =5\right)^{-n +3}}{3857}+\frac{3919 \mathit{RootOf} \left(Z^{5}-7 Z^{4}+18 Z^{3}-17 Z^{2}+7 Z -1, \mathit{index} =1\right)^{-n +2}}{3857}+\frac{3919 \mathit{RootOf} \left(Z^{5}-7 Z^{4}+18 Z^{3}-17 Z^{2}+7 Z -1, \mathit{index} =2\right)^{-n +2}}{3857}+\frac{3919 \mathit{RootOf} \left(Z^{5}-7 Z^{4}+18 Z^{3}-17 Z^{2}+7 Z -1, \mathit{index} =3\right)^{-n +2}}{3857}+\frac{3919 \mathit{RootOf} \left(Z^{5}-7 Z^{4}+18 Z^{3}-17 Z^{2}+7 Z -1, \mathit{index} =4\right)^{-n +2}}{3857}+\frac{3919 \mathit{RootOf} \left(Z^{5}-7 Z^{4}+18 Z^{3}-17 Z^{2}+7 Z -1, \mathit{index} =5\right)^{-n +2}}{3857}-\frac{8368 \mathit{RootOf} \left(Z^{5}-7 Z^{4}+18 Z^{3}-17 Z^{2}+7 Z -1, \mathit{index} =1\right)^{-n +1}}{3857}-\frac{8368 \mathit{RootOf} \left(Z^{5}-7 Z^{4}+18 Z^{3}-17 Z^{2}+7 Z -1, \mathit{index} =2\right)^{-n +1}}{3857}-\frac{8368 \mathit{RootOf} \left(Z^{5}-7 Z^{4}+18 Z^{3}-17 Z^{2}+7 Z -1, \mathit{index} =3\right)^{-n +1}}{3857}-\frac{8368 \mathit{RootOf} \left(Z^{5}-7 Z^{4}+18 Z^{3}-17 Z^{2}+7 Z -1, \mathit{index} =4\right)^{-n +1}}{3857}-\frac{8368 \mathit{RootOf} \left(Z^{5}-7 Z^{4}+18 Z^{3}-17 Z^{2}+7 Z -1, \mathit{index} =5\right)^{-n +1}}{3857}-\frac{737 \mathit{RootOf} \left(Z^{5}-7 Z^{4}+18 Z^{3}-17 Z^{2}+7 Z -1, \mathit{index} =1\right)^{-n -1}}{3857}-\frac{737 \mathit{RootOf} \left(Z^{5}-7 Z^{4}+18 Z^{3}-17 Z^{2}+7 Z -1, \mathit{index} =2\right)^{-n -1}}{3857}-\frac{737 \mathit{RootOf} \left(Z^{5}-7 Z^{4}+18 Z^{3}-17 Z^{2}+7 Z -1, \mathit{index} =3\right)^{-n -1}}{3857}-\frac{737 \mathit{RootOf} \left(Z^{5}-7 Z^{4}+18 Z^{3}-17 Z^{2}+7 Z -1, \mathit{index} =4\right)^{-n -1}}{3857}-\frac{737 \mathit{RootOf} \left(Z^{5}-7 Z^{4}+18 Z^{3}-17 Z^{2}+7 Z -1, \mathit{index} =5\right)^{-n -1}}{3857}+\frac{283 \mathit{RootOf} \left(Z^{5}-7 Z^{4}+18 Z^{3}-17 Z^{2}+7 Z -1, \mathit{index} =1\right)^{-n}}{203}+\frac{283 \mathit{RootOf} \left(Z^{5}-7 Z^{4}+18 Z^{3}-17 Z^{2}+7 Z -1, \mathit{index} =2\right)^{-n}}{203}+\frac{283 \mathit{RootOf} \left(Z^{5}-7 Z^{4}+18 Z^{3}-17 Z^{2}+7 Z -1, \mathit{index} =3\right)^{-n}}{203}+\frac{283 \mathit{RootOf} \left(Z^{5}-7 Z^{4}+18 Z^{3}-17 Z^{2}+7 Z -1, \mathit{index} =4\right)^{-n}}{203}+\frac{283 \mathit{RootOf} \left(Z^{5}-7 Z^{4}+18 Z^{3}-17 Z^{2}+7 Z -1, \mathit{index} =5\right)^{-n}}{203}\)
This specification was found using the strategy pack "Point Placements" and has 47 rules.
Found on January 18, 2022.Finding the specification took 1 seconds.
Copy 47 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_{5}\! \left(x \right)\\
F_{4}\! \left(x \right) &= x\\
F_{5}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{8}\! \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_{6}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{18}\! \left(x \right)+F_{33}\! \left(x \right)\\
F_{17}\! \left(x \right) &= 0\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{21}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{26}\! \left(x \right)\\
F_{30}\! \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_{20}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= 2 F_{17}\! \left(x \right)+F_{31}\! \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 \right)+F_{43}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{4}\! \left(x \right) F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{39}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{4}\! \left(x \right) F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{43}\! \left(x \right)\\
\end{align*}\)