Av(1243, 1342, 1432, 2413, 3124)
Generating Function
\(\displaystyle -\frac{\left(2 x -1\right) \left(x -1\right)^{3}}{\left(x^{2}-x +1\right) \left(2 x^{4}+2 x^{3}-7 x^{2}+5 x -1\right)}\)
Counting Sequence
1, 1, 2, 6, 19, 57, 167, 487, 1420, 4140, 12067, 35167, 102485, 298671, 870430, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{2}-x +1\right) \left(2 x^{4}+2 x^{3}-7 x^{2}+5 x -1\right) F \! \left(x \right)+\left(2 x -1\right) \left(x -1\right)^{3} = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 57\)
\(\displaystyle a \! \left(n \right) = \frac{7 a \! \left(n +2\right)}{2}-7 a \! \left(n +3\right)+\frac{13 a \! \left(n +4\right)}{2}-3 a \! \left(n +5\right)+\frac{a \! \left(n +6\right)}{2}, \quad n \geq 6\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 57\)
\(\displaystyle a \! \left(n \right) = \frac{7 a \! \left(n +2\right)}{2}-7 a \! \left(n +3\right)+\frac{13 a \! \left(n +4\right)}{2}-3 a \! \left(n +5\right)+\frac{a \! \left(n +6\right)}{2}, \quad n \geq 6\)
Explicit Closed Form
\(\displaystyle \frac{99 \left(\underset{\alpha =\mathit{RootOf} \left(2 Z^{6}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +4}\right)}{596}-\frac{13 \left(\underset{\alpha =\mathit{RootOf} \left(2 Z^{6}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +3}\right)}{298}-\frac{875 \left(\underset{\alpha =\mathit{RootOf} \left(2 Z^{6}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +2}\right)}{1192}+\frac{689 \left(\underset{\alpha =\mathit{RootOf} \left(2 Z^{6}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +1}\right)}{596}-\frac{467 \left(\underset{\alpha =\mathit{RootOf} \left(2 Z^{6}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n}\right)}{596}+\frac{305 \left(\underset{\alpha =\mathit{RootOf} \left(2 Z^{6}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n -1}\right)}{1192}\)
This specification was found using the strategy pack "Point Placements" and has 60 rules.
Found on January 18, 2022.Finding the specification took 1 seconds.
Copy 60 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_{17}\! \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_{13}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{10}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{20}\! \left(x \right)+F_{39}\! \left(x \right)\\
F_{19}\! \left(x \right) &= 0\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= 2 F_{19}\! \left(x \right)+F_{29}\! \left(x \right)+F_{33}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{23}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{28}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{36}\! \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_{2}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{4}\! \left(x \right) F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{43}\! \left(x \right)+F_{55}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{4}\! \left(x \right) F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{48}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{26}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= 2 F_{19}\! \left(x \right)+F_{50}\! \left(x \right)+F_{54}\! \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_{53}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{46}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{49}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{35}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{4}\! \left(x \right) F_{41}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{4}\! \left(x \right) F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{42}\! \left(x \right)\\
\end{align*}\)