Av(1342, 1432, 2314, 4123)
Generating Function
\(\displaystyle -\frac{\left(2 x -1\right) \left(x^{8}+x^{7}-x^{5}-4 x^{4}-x^{3}-2 x^{2}+3 x -1\right)}{\left(x^{2}-3 x +1\right) \left(x^{3}+x^{2}+x -1\right) \left(x -1\right)^{2}}\)
Counting Sequence
1, 1, 2, 6, 20, 60, 166, 442, 1158, 3018, 7858, 20472, 53388, 139362, 364072, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{2}-3 x +1\right) \left(x^{3}+x^{2}+x -1\right) \left(x -1\right)^{2} F \! \left(x \right)+\left(2 x -1\right) \left(x^{8}+x^{7}-x^{5}-4 x^{4}-x^{3}-2 x^{2}+3 x -1\right) = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 60\)
\(\displaystyle a \! \left(6\right) = 166\)
\(\displaystyle a \! \left(7\right) = 442\)
\(\displaystyle a \! \left(8\right) = 1158\)
\(\displaystyle a \! \left(9\right) = 3018\)
\(\displaystyle a \! \left(n +5\right) = a \! \left(n \right)-2 a \! \left(n +1\right)-a \! \left(n +2\right)-3 a \! \left(n +3\right)+4 a \! \left(n +4\right)-4 n -6, \quad n \geq 10\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 60\)
\(\displaystyle a \! \left(6\right) = 166\)
\(\displaystyle a \! \left(7\right) = 442\)
\(\displaystyle a \! \left(8\right) = 1158\)
\(\displaystyle a \! \left(9\right) = 3018\)
\(\displaystyle a \! \left(n +5\right) = a \! \left(n \right)-2 a \! \left(n +1\right)-a \! \left(n +2\right)-3 a \! \left(n +3\right)+4 a \! \left(n +4\right)-4 n -6, \quad n \geq 10\)
Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0 \\ 1 & n =1 \\ 2 & n =2 \\ \frac{\left(\left(\left(80 \,\mathrm{I}-20 \sqrt{11}\right) \sqrt{3}-60 \,\mathrm{I} \sqrt{11}+80\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}+40+\left(\left(155 \,\mathrm{I}+25 \sqrt{11}\right) \sqrt{3}-75 \,\mathrm{I} \sqrt{11}-155\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}}{1320}\\+\\\frac{\left(\left(\left(25 \sqrt{11}-155 \,\mathrm{I}\right) \sqrt{3}+75 \,\mathrm{I} \sqrt{11}-155\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}+40+\left(\left(-20 \sqrt{11}-80 \,\mathrm{I}\right) \sqrt{3}+60 \,\mathrm{I} \sqrt{11}+80\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}}{1320}\\+\\\frac{\left(\left(-50 \sqrt{11}\, \sqrt{3}+310\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}+40 \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}} \sqrt{11}\, \sqrt{3}-160 \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}+40\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}}{1320}\\+\frac{\left(-8232 \sqrt{5}+19080\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{1320}+\frac{\left(8232 \sqrt{5}+19080\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{1320}\\-2 n -1 & \text{otherwise} \end{array}\right.\)
This specification was found using the strategy pack "Point Placements" and has 103 rules.
Found on January 18, 2022.Finding the specification took 1 seconds.
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\(\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_{19}\! \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_{6}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \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_{17}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{4}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{26}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{13}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{29}\! \left(x \right)+F_{57}\! \left(x \right)\\
F_{28}\! \left(x \right) &= 0\\
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_{35}\! \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_{4}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= 2 F_{28}\! \left(x \right)+F_{37}\! \left(x \right)+F_{41}\! \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_{40}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{32}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{36}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{4}\! \left(x \right) F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{44}\! \left(x \right)+F_{56}\! \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_{49}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= x^{2}\\
F_{49}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= 2 F_{28}\! \left(x \right)+F_{51}\! \left(x \right)+F_{55}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{4}\! \left(x \right) F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)+F_{54}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{47}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{50}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{4}\! \left(x \right) F_{43}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{20}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{4}\! \left(x \right) F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{60}\! \left(x \right)+F_{98}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{4}\! \left(x \right) F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{62}\! \left(x \right)+F_{70}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{63}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{64}\! \left(x \right)+F_{65}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{16}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{4}\! \left(x \right) F_{66}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{67}\! \left(x \right)+F_{68}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{4}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{69}\! \left(x \right)\\
F_{69}\! \left(x \right) &= x^{2}\\
F_{70}\! \left(x \right) &= F_{59}\! \left(x \right)+F_{71}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{72}\! \left(x \right)+F_{76}\! \left(x \right)+F_{93}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{4}\! \left(x \right) F_{73}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{74}\! \left(x \right)+F_{75}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{63}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{71}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{4}\! \left(x \right) F_{77}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{78}\! \left(x \right)+F_{92}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{79}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{4}\! \left(x \right) F_{80}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{81}\! \left(x \right)+F_{82}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{78}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{83}\! \left(x \right)+F_{87}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{84}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{4}\! \left(x \right) F_{85}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{86}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{56}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{88}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{4}\! \left(x \right) F_{89}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{78}\! \left(x \right)+F_{90}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{91}\! \left(x \right)\\
F_{91}\! \left(x \right) &= F_{4}\! \left(x \right) F_{78}\! \left(x \right)\\
F_{92}\! \left(x \right) &= F_{91}\! \left(x \right)\\
F_{93}\! \left(x \right) &= F_{4}\! \left(x \right) F_{94}\! \left(x \right)\\
F_{94}\! \left(x \right) &= F_{95}\! \left(x \right)+F_{96}\! \left(x \right)\\
F_{95}\! \left(x \right) &= F_{43}\! \left(x \right)\\
F_{96}\! \left(x \right) &= F_{97}\! \left(x \right)\\
F_{97}\! \left(x \right) &= F_{4}\! \left(x \right) F_{86}\! \left(x \right)\\
F_{98}\! \left(x \right) &= F_{4}\! \left(x \right) F_{99}\! \left(x \right)\\
F_{99}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{81}\! \left(x \right)\\
F_{100}\! \left(x \right) &= F_{101}\! \left(x \right)+F_{102}\! \left(x \right)\\
F_{101}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{44}\! \left(x \right)+F_{84}\! \left(x \right)\\
F_{102}\! \left(x \right) &= F_{88}\! \left(x \right)\\
\end{align*}\)