Av(1234, 1243, 2431, 3214, 3241)
Generating Function
\(\displaystyle -\frac{3 x^{8}+4 x^{7}-2 x^{6}-11 x^{5}-8 x^{4}-2 x^{3}+x^{2}+x -1}{\left(x^{2}+x -1\right) \left(x^{3}+x^{2}+x -1\right)}\)
Counting Sequence
1, 1, 2, 6, 19, 50, 110, 233, 479, 962, 1900, 3707, 7161, 13726, 26144, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{2}+x -1\right) \left(x^{3}+x^{2}+x -1\right) F \! \left(x \right)+3 x^{8}+4 x^{7}-2 x^{6}-11 x^{5}-8 x^{4}-2 x^{3}+x^{2}+x -1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 50\)
\(\displaystyle a \! \left(6\right) = 110\)
\(\displaystyle a \! \left(7\right) = 233\)
\(\displaystyle a \! \left(8\right) = 479\)
\(\displaystyle a \! \left(n +5\right) = -a \! \left(n \right)-2 a \! \left(n +1\right)-a \! \left(n +2\right)+a \! \left(n +3\right)+2 a \! \left(n +4\right), \quad n \geq 9\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 50\)
\(\displaystyle a \! \left(6\right) = 110\)
\(\displaystyle a \! \left(7\right) = 233\)
\(\displaystyle a \! \left(8\right) = 479\)
\(\displaystyle a \! \left(n +5\right) = -a \! \left(n \right)-2 a \! \left(n +1\right)-a \! \left(n +2\right)+a \! \left(n +3\right)+2 a \! \left(n +4\right), \quad n \geq 9\)
Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0 \\ 1 & n =1 \\ 2 & n =2 \\ 6 & n =3 \\ \frac{\left(\left(\left(180 \,\mathrm{I}-40 \sqrt{11}\right) \sqrt{3}-120 \,\mathrm{I} \sqrt{11}+180\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}+720+\left(\left(225 \,\mathrm{I}+35 \sqrt{11}\right) \sqrt{3}-105 \,\mathrm{I} \sqrt{11}-225\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}\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}}{240}\\+\\\frac{\left(\left(\left(-225 \,\mathrm{I}+35 \sqrt{11}\right) \sqrt{3}+105 \,\mathrm{I} \sqrt{11}-225\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}+720+\left(\left(-180 \,\mathrm{I}-40 \sqrt{11}\right) \sqrt{3}+120 \,\mathrm{I} \sqrt{11}+180\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}\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}}{240}\\+\\\frac{\left(\left(-70 \sqrt{11}\, \sqrt{3}+450\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}+80 \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}} \sqrt{11}\, \sqrt{3}-360 \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}+720\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}}{240}\\+\frac{\left(192 \sqrt{5}-1440\right) \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{240}+\frac{\left(-192 \sqrt{5}-1440\right) \left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{240} & \text{otherwise} \end{array}\right.\)
This specification was found using the strategy pack "Point Placements" and has 47 rules.
Found on January 18, 2022.Finding the specification took 1 seconds.
Copy 47 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_{11}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{14}\! \left(x \right)+F_{16}\! \left(x \right)\\
F_{13}\! \left(x \right) &= 0\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{4}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{4}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{19}\! \left(x \right)+F_{29}\! \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_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{14}\! \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_{11}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{14}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{26}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{33}\! \left(x \right)+F_{36}\! \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_{35}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{2}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{4}\! \left(x \right) F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{42}\! \left(x \right) &= 2 F_{13}\! \left(x \right)+F_{43}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{4}\! \left(x \right) F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{38}\! \left(x \right) F_{4}\! \left(x \right)\\
\end{align*}\)