Av(1243, 2314, 2413, 2431, 3124)
Generating Function
\(\displaystyle -\frac{2 x^{6}-6 x^{5}+12 x^{4}-16 x^{3}+14 x^{2}-6 x +1}{\left(x -1\right) \left(x^{3}-2 x^{2}+3 x -1\right)^{2}}\)
Counting Sequence
1, 1, 2, 6, 19, 57, 163, 451, 1218, 3228, 8428, 21742, 55542, 140740, 354197, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x -1\right) \left(x^{3}-2 x^{2}+3 x -1\right)^{2} F \! \left(x \right)+2 x^{6}-6 x^{5}+12 x^{4}-16 x^{3}+14 x^{2}-6 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 57\)
\(\displaystyle a \! \left(6\right) = 163\)
\(\displaystyle a \! \left(n +6\right) = -a \! \left(n \right)+4 a \! \left(n +1\right)-10 a \! \left(n +2\right)+14 a \! \left(n +3\right)-13 a \! \left(n +4\right)+6 a \! \left(n +5\right)+1, \quad n \geq 7\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 57\)
\(\displaystyle a \! \left(6\right) = 163\)
\(\displaystyle a \! \left(n +6\right) = -a \! \left(n \right)+4 a \! \left(n +1\right)-10 a \! \left(n +2\right)+14 a \! \left(n +3\right)-13 a \! \left(n +4\right)+6 a \! \left(n +5\right)+1, \quad n \geq 7\)
Explicit Closed Form
\(\displaystyle \frac{\left(-3105 \,2^{\frac{2}{3}} \left(-\frac{\left(\mathrm{I}-\frac{\sqrt{3}}{3}\right) \left(n -\frac{104}{23}\right) \sqrt{23}}{9}+\left(\mathrm{I} \sqrt{3}-1\right) \left(n -\frac{22}{9}\right)\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}+207 \,2^{\frac{1}{3}} \left(-\frac{7 \left(\mathrm{I}+\frac{\sqrt{3}}{3}\right) \left(n -\frac{341}{161}\right) \sqrt{23}}{3}+\left(1+\mathrm{I} \sqrt{3}\right) \left(n +\frac{7}{3}\right)\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}+55200 n \right) \left(\frac{11 \,2^{\frac{1}{3}} \left(\left(\mathrm{I}-\frac{3 \sqrt{23}}{11}\right) \sqrt{3}-\frac{9 \,\mathrm{I} \sqrt{23}}{11}+1\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}}{600}-\frac{\mathrm{I} \sqrt{3}\, \left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{\left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{2}{3}\right)^{-n}}{317400}+\frac{\left(3105 \,2^{\frac{2}{3}} \left(-\frac{\left(\mathrm{I}+\frac{\sqrt{3}}{3}\right) \left(n -\frac{104}{23}\right) \sqrt{23}}{9}+\left(n -\frac{22}{9}\right) \left(1+\mathrm{I} \sqrt{3}\right)\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}-207 \left(-\frac{7 \left(\mathrm{I}-\frac{\sqrt{3}}{3}\right) \left(n -\frac{341}{161}\right) \sqrt{23}}{3}+\left(\mathrm{I} \sqrt{3}-1\right) \left(n +\frac{7}{3}\right)\right) 2^{\frac{1}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}+55200 n \right) \left(-\frac{11 \,2^{\frac{1}{3}} \left(\left(\mathrm{I}+\frac{3 \sqrt{23}}{11}\right) \sqrt{3}-\frac{9 \,\mathrm{I} \sqrt{23}}{11}-1\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}}{600}+\frac{\mathrm{I} \sqrt{3}\, \left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{\left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{2}{3}\right)^{-n}}{317400}+1+\frac{\left(230 \,2^{\frac{2}{3}} \left(\sqrt{3}\, \left(n -\frac{104}{23}\right) \sqrt{23}-27 n +66\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}+322 \,2^{\frac{1}{3}} \left(\sqrt{3}\, \left(n -\frac{341}{161}\right) \sqrt{23}-\frac{9 n}{7}-3\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}+55200 n \right) \left(\frac{2^{\frac{1}{3}} \left(3 \sqrt{23}\, \sqrt{3}-11\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}}{300}-\frac{\left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}+\frac{2}{3}\right)^{-n}}{317400}\)
This specification was found using the strategy pack "Point Placements" and has 42 rules.
Found on January 18, 2022.Finding the specification took 1 seconds.
Copy 42 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_{23}\! \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_{16}\! \left(x \right)+F_{20}\! \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_{18}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{14}\! \left(x \right)\\
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_{7}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{20}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{25}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{21}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{24}\! \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_{35}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{28}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{32}\! \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_{39}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{39}\! \left(x \right) F_{4}\! \left(x \right)\\
\end{align*}\)