Av(1243, 1324, 2341, 2431, 4213)
Generating Function
\(\displaystyle -\frac{2 x^{9}+6 x^{8}+x^{7}-7 x^{6}-2 x^{5}-3 x^{4}+2 x^{3}+5 x^{2}-4 x +1}{\left(x^{2}+x -1\right)^{2} \left(x -1\right)^{3}}\)
Counting Sequence
1, 1, 2, 6, 19, 51, 115, 234, 447, 821, 1470, 2588, 4504, 7775, 13342, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{2}+x -1\right)^{2} \left(x -1\right)^{3} F \! \left(x \right)+2 x^{9}+6 x^{8}+x^{7}-7 x^{6}-2 x^{5}-3 x^{4}+2 x^{3}+5 x^{2}-4 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 51\)
\(\displaystyle a \! \left(6\right) = 115\)
\(\displaystyle a \! \left(7\right) = 234\)
\(\displaystyle a \! \left(8\right) = 447\)
\(\displaystyle a \! \left(9\right) = 821\)
\(\displaystyle a \! \left(n +4\right) = \frac{n^{2}}{2}-a \! \left(n \right)-2 a \! \left(n +1\right)+a \! \left(n +2\right)+2 a \! \left(n +3\right)-\frac{31 n}{2}+39, \quad n \geq 10\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 51\)
\(\displaystyle a \! \left(6\right) = 115\)
\(\displaystyle a \! \left(7\right) = 234\)
\(\displaystyle a \! \left(8\right) = 447\)
\(\displaystyle a \! \left(9\right) = 821\)
\(\displaystyle a \! \left(n +4\right) = \frac{n^{2}}{2}-a \! \left(n \right)-2 a \! \left(n +1\right)+a \! \left(n +2\right)+2 a \! \left(n +3\right)-\frac{31 n}{2}+39, \quad n \geq 10\)
Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ \frac{\left(\left(-10 n -46\right) \sqrt{5}+20 n +100\right) \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{50}+\\\frac{\left(\left(10 n +46\right) \sqrt{5}+20 n +100\right) \left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{50}+\frac{n^{2}}{2}-\frac{27 n}{2}+14 & \text{otherwise} \end{array}\right.\)
This specification was found using the strategy pack "Point Placements" and has 67 rules.
Found on January 18, 2022.Finding the specification took 1 seconds.
Copy 67 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_{14}\! \left(x \right)+F_{7}\! \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_{7}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{27}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{26}\! \left(x \right) &= 0\\
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_{43}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{31}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{33}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{14}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{39}\! \left(x \right)\\
F_{39}\! \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_{33}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{25}\! \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_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{4}\! \left(x \right) F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{41}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{53}\! \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_{47}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{4}\! \left(x \right) F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{60}\! \left(x \right)+F_{64}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{4}\! \left(x \right) F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{63}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{4}\! \left(x \right) F_{66}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{14}\! \left(x \right)\\
\end{align*}\)