Av(1243, 2314, 2341, 2431, 3214)
Generating Function
\(\displaystyle -\frac{x^{9}+2 x^{8}+2 x^{7}+3 x^{6}+x^{5}-5 x^{4}-2 x^{3}-2 x^{2}+3 x -1}{\left(x^{3}+x^{2}+x -1\right)^{2} \left(x -1\right)^{2}}\)
Counting Sequence
1, 1, 2, 6, 19, 53, 133, 314, 707, 1540, 3276, 6842, 14087, 28675, 57827, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{3}+x^{2}+x -1\right)^{2} \left(x -1\right)^{2} F \! \left(x \right)+x^{9}+2 x^{8}+2 x^{7}+3 x^{6}+x^{5}-5 x^{4}-2 x^{3}-2 x^{2}+3 x -1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 53\)
\(\displaystyle a \! \left(6\right) = 133\)
\(\displaystyle a \! \left(7\right) = 314\)
\(\displaystyle a \! \left(8\right) = 707\)
\(\displaystyle a \! \left(9\right) = 1540\)
\(\displaystyle a \! \left(n +2\right) = -\frac{a \! \left(n \right)}{3}-\frac{2 a \! \left(n +1\right)}{3}+\frac{a \! \left(n +4\right)}{3}+\frac{2 a \! \left(n +5\right)}{3}-\frac{a \! \left(n +6\right)}{3}-\frac{2 n}{3}+7, \quad n \geq 10\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 53\)
\(\displaystyle a \! \left(6\right) = 133\)
\(\displaystyle a \! \left(7\right) = 314\)
\(\displaystyle a \! \left(8\right) = 707\)
\(\displaystyle a \! \left(9\right) = 1540\)
\(\displaystyle a \! \left(n +2\right) = -\frac{a \! \left(n \right)}{3}-\frac{2 a \! \left(n +1\right)}{3}+\frac{a \! \left(n +4\right)}{3}+\frac{2 a \! \left(n +5\right)}{3}-\frac{a \! \left(n +6\right)}{3}-\frac{2 n}{3}+7, \quad n \geq 10\)
Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0 \\ 1 & n =1 \\ \frac{\left(\left(-1254 \left(\mathrm{I}+\frac{\sqrt{3}}{3}\right) \left(n -\frac{544}{209}\right) \sqrt{11}+1914 \left(1+\mathrm{I} \sqrt{3}\right) \left(n -\frac{76}{29}\right)\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}+\left(-1023 \left(\mathrm{I}-\frac{\sqrt{3}}{3}\right) \left(n -\frac{862}{341}\right) \sqrt{11}+2211 \left(\mathrm{I} \sqrt{3}-1\right) \left(n -\frac{170}{67}\right)\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}+3168 n -8712\right) \left(\frac{\left(\left(17 \,\mathrm{I}+3 \sqrt{11}\right) \sqrt{3}-9 \,\mathrm{I} \sqrt{11}-17\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{24}-\frac{\mathrm{I} \sqrt{3}\, \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{1}{3}\right)^{-n}}{11616}\\+\\\frac{\left(\left(1254 \left(\mathrm{I}-\frac{\sqrt{3}}{3}\right) \left(n -\frac{544}{209}\right) \sqrt{11}-1914 \left(\mathrm{I} \sqrt{3}-1\right) \left(n -\frac{76}{29}\right)\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}+\left(1023 \left(\mathrm{I}+\frac{\sqrt{3}}{3}\right) \left(n -\frac{862}{341}\right) \sqrt{11}-2211 \left(1+\mathrm{I} \sqrt{3}\right) \left(n -\frac{170}{67}\right)\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}+3168 n -8712\right) \left(\frac{\left(\left(-17 \,\mathrm{I}+3 \sqrt{11}\right) \sqrt{3}+9 \,\mathrm{I} \sqrt{11}-17\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{24}+\frac{\mathrm{I} \sqrt{3}\, \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{1}{3}\right)^{-n}}{11616}\\+\\\frac{\left(\left(\left(836 n -2176\right) \sqrt{3}\, \sqrt{11}-3828 n +10032\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}+\left(\left(-682 n +1724\right) \sqrt{3}\, \sqrt{11}+4422 n -11220\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}+3168 n -8712\right) \left(\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{3}-\frac{1}{3}+\frac{17 \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{12}-\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{11}\, \sqrt{3}}{4}\right)^{-n}}{11616}\\-\frac{n}{2}+\frac{21}{4} & \text{otherwise} \end{array}\right.\)
This specification was found using the strategy pack "Point Placements" and has 50 rules.
Found on January 18, 2022.Finding the specification took 1 seconds.
Copy 50 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_{24}\! \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_{22}\! \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_{21}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{18}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{26}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{23}\! \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_{34}\! \left(x \right) &= F_{35}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{4}\! \left(x \right) F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{4}\! \left(x \right) F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{4}\! \left(x \right) F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{44}\! \left(x \right)\\
\end{align*}\)