Av(1432, 2341, 2413, 3214)
Generating Function
\(\displaystyle \frac{2 x^{9}-7 x^{8}+10 x^{7}-6 x^{6}-5 x^{5}+7 x^{4}-7 x^{3}+9 x^{2}-5 x +1}{\left(x^{3}-2 x^{2}+3 x -1\right) \left(2 x^{4}+2 x^{3}+x^{2}+x -1\right) \left(x -1\right)^{2}}\)
Counting Sequence
1, 1, 2, 6, 20, 59, 160, 423, 1094, 2774, 6935, 17158, 42087, 102503, 248189, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{3}-2 x^{2}+3 x -1\right) \left(2 x^{4}+2 x^{3}+x^{2}+x -1\right) \left(x -1\right)^{2} F \! \left(x \right)-2 x^{9}+7 x^{8}-10 x^{7}+6 x^{6}+5 x^{5}-7 x^{4}+7 x^{3}-9 x^{2}+5 x -1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 59\)
\(\displaystyle a \! \left(6\right) = 160\)
\(\displaystyle a \! \left(7\right) = 423\)
\(\displaystyle a \! \left(8\right) = 1094\)
\(\displaystyle a \! \left(9\right) = 2774\)
\(\displaystyle a \! \left(n +7\right) = -2 a \! \left(n \right)+2 a \! \left(n +1\right)-3 a \! \left(n +2\right)-3 a \! \left(n +3\right)+2 a \! \left(n +4\right)-4 a \! \left(n +5\right)+4 a \! \left(n +6\right)-n +1, \quad n \geq 10\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 59\)
\(\displaystyle a \! \left(6\right) = 160\)
\(\displaystyle a \! \left(7\right) = 423\)
\(\displaystyle a \! \left(8\right) = 1094\)
\(\displaystyle a \! \left(9\right) = 2774\)
\(\displaystyle a \! \left(n +7\right) = -2 a \! \left(n \right)+2 a \! \left(n +1\right)-3 a \! \left(n +2\right)-3 a \! \left(n +3\right)+2 a \! \left(n +4\right)-4 a \! \left(n +5\right)+4 a \! \left(n +6\right)-n +1, \quad n \geq 10\)
Explicit Closed Form
\(\displaystyle \frac{13}{25}-\frac{n}{5}-\frac{2051124074 \left(\underset{\alpha =\mathit{RootOf} \left(2 Z^{7}-2 Z^{6}+3 Z^{5}+3 Z^{4}-2 Z^{3}+4 Z^{2}-4 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +5}\right)}{125160825}+\frac{226160972 \left(\underset{\alpha =\mathit{RootOf} \left(2 Z^{7}-2 Z^{6}+3 Z^{5}+3 Z^{4}-2 Z^{3}+4 Z^{2}-4 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +4}\right)}{25032165}-\frac{2576057126 \left(\underset{\alpha =\mathit{RootOf} \left(2 Z^{7}-2 Z^{6}+3 Z^{5}+3 Z^{4}-2 Z^{3}+4 Z^{2}-4 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +3}\right)}{125160825}-\frac{4234551847 \left(\underset{\alpha =\mathit{RootOf} \left(2 Z^{7}-2 Z^{6}+3 Z^{5}+3 Z^{4}-2 Z^{3}+4 Z^{2}-4 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +2}\right)}{125160825}+\frac{154568057 \left(\underset{\alpha =\mathit{RootOf} \left(2 Z^{7}-2 Z^{6}+3 Z^{5}+3 Z^{4}-2 Z^{3}+4 Z^{2}-4 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +1}\right)}{125160825}-\frac{4058673121 \left(\underset{\alpha =\mathit{RootOf} \left(2 Z^{7}-2 Z^{6}+3 Z^{5}+3 Z^{4}-2 Z^{3}+4 Z^{2}-4 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n}\right)}{125160825}+\frac{2279861467 \left(\underset{\alpha =\mathit{RootOf} \left(2 Z^{7}-2 Z^{6}+3 Z^{5}+3 Z^{4}-2 Z^{3}+4 Z^{2}-4 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n -1}\right)}{125160825}+\left(\left\{\begin{array}{cc}1 & n =0 \\ 0 & \text{otherwise} \end{array}\right.\right)\)
This specification was found using the strategy pack "Point Placements" and has 57 rules.
Found on January 18, 2022.Finding the specification took 1 seconds.
Copy 57 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_{24}\! \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_{14}\! \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_{1}\! \left(x \right)+F_{11}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{15}\! \left(x \right) &= 0\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{21}\! \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_{21}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{26}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{18}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{36}\! \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_{40}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{2}\! \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_{2}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{4}\! \left(x \right) F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{41}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{4}\! \left(x \right) F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{4}\! \left(x \right) F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{4}\! \left(x \right) F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{52}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{41}\! \left(x \right)\\
\end{align*}\)