Av(1243, 1324, 2341, 3142)
Generating Function
\(\displaystyle \frac{\left(2 x -1\right) \left(2 x^{4}-7 x^{2}+5 x -1\right)}{\left(x -1\right) \left(x^{2}+x -1\right) \left(x^{2}-3 x +1\right)^{2}}\)
Counting Sequence
1, 1, 2, 6, 20, 64, 197, 589, 1726, 4986, 14252, 40408, 113825, 318921, 889538, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x -1\right) \left(x^{2}+x -1\right) \left(x^{2}-3 x +1\right)^{2} F \! \left(x \right)-\left(2 x -1\right) \left(2 x^{4}-7 x^{2}+5 x -1\right) = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 64\)
\(\displaystyle a \! \left(n +6\right) = a \! \left(n \right)-5 a \! \left(n +1\right)+4 a \! \left(n +2\right)+11 a \! \left(n +3\right)-16 a \! \left(n +4\right)+7 a \! \left(n +5\right)-1, \quad n \geq 6\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 64\)
\(\displaystyle a \! \left(n +6\right) = a \! \left(n \right)-5 a \! \left(n +1\right)+4 a \! \left(n +2\right)+11 a \! \left(n +3\right)-16 a \! \left(n +4\right)+7 a \! \left(n +5\right)-1, \quad n \geq 6\)
Explicit Closed Form
\(\displaystyle -\frac{\left(3+\sqrt{5}\right) \left(\left(\left(3 n +\frac{17}{10}\right) \sqrt{5}-7 n -\frac{9}{2}\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}+\left(-2 n +\frac{4 \sqrt{5}}{5}-3\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}+\left(-\frac{7 \sqrt{5}}{2}+\frac{15}{2}\right) \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}-15+\left(\left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}+5\right) \sqrt{5}\right)}{20}\)
This specification was found using the strategy pack "Point Placements" and has 40 rules.
Found on July 23, 2021.Finding the specification took 8 seconds.
Copy 40 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_{0}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{4}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{15}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{0}\! \left(x \right) F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{13}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{12}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right) F_{20}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{12}\! \left(x \right) F_{17}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{17}\! \left(x \right) F_{23}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{21}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{13} \left(x \right)^{2}\\
F_{25}\! \left(x \right) &= F_{12}\! \left(x \right) F_{6}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right) F_{31}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{12}\! \left(x \right) F_{21}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{12}\! \left(x \right) F_{35}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{35}\! \left(x \right) F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)\\
\end{align*}\)