Av(1243, 2431, 4132, 4231)
Generating Function
\(\displaystyle -\frac{x^{8}+4 x^{7}-40 x^{6}+97 x^{5}-125 x^{4}+95 x^{3}-42 x^{2}+10 x -1}{\left(x^{2}-3 x +1\right) \left(2 x -1\right)^{2} \left(x -1\right)^{4}}\)
Counting Sequence
1, 1, 2, 6, 20, 65, 199, 577, 1607, 4355, 11596, 30542, 79927, 208422, 542539, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{2}-3 x +1\right) \left(2 x -1\right)^{2} \left(x -1\right)^{4} F \! \left(x \right)+x^{8}+4 x^{7}-40 x^{6}+97 x^{5}-125 x^{4}+95 x^{3}-42 x^{2}+10 x -1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 65\)
\(\displaystyle a \! \left(6\right) = 199\)
\(\displaystyle a \! \left(7\right) = 577\)
\(\displaystyle a \! \left(8\right) = 1607\)
\(\displaystyle a \! \left(n +4\right) = \frac{n^{3}}{6}-n^{2}-4 a \! \left(n \right)+16 a \! \left(n +1\right)-17 a \! \left(n +2\right)+7 a \! \left(n +3\right)-\frac{7 n}{6}+1, \quad n \geq 9\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 65\)
\(\displaystyle a \! \left(6\right) = 199\)
\(\displaystyle a \! \left(7\right) = 577\)
\(\displaystyle a \! \left(8\right) = 1607\)
\(\displaystyle a \! \left(n +4\right) = \frac{n^{3}}{6}-n^{2}-4 a \! \left(n \right)+16 a \! \left(n +1\right)-17 a \! \left(n +2\right)+7 a \! \left(n +3\right)-\frac{7 n}{6}+1, \quad n \geq 9\)
Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(6 \sqrt{5}+30\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{60}+\frac{\left(-6 \sqrt{5}+30\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{60}+\frac{\left(15 n -105\right) 2^{n}}{60}\\-\frac{n^{3}}{6}+\frac{n^{2}}{2}-\frac{n}{3}+2 & \text{otherwise} \end{array}\right.\)
This specification was found using the strategy pack "Insertion Point Placements" and has 62 rules.
Found on January 17, 2022.Finding the specification took 6 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_{4}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{22}\! \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_{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_{8}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{21}\! \left(x \right)\\
F_{18}\! \left(x \right) &= 0\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{13}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{39}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{11}\! \left(x \right) F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{36}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{34}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{16}\! \left(x \right) F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{21}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{10}\! \left(x \right) F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{43}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{45}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{11}\! \left(x \right) F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{48}\! \left(x \right)+F_{50}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{10}\! \left(x \right) F_{47}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{47}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{42}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{53}\! \left(x \right) F_{6}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)+F_{57}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{33}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{44}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{14}\! \left(x \right) F_{55}\! \left(x \right) F_{61}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{5}\! \left(x \right)+F_{6}\! \left(x \right)\\
\end{align*}\)