Av(1243, 1342, 1432, 2341, 2413)
Generating Function
\(\displaystyle \frac{x^{4}-2 x^{3}+5 x^{2}-4 x +1}{\left(x -1\right) \left(2 x^{3}-4 x^{2}+4 x -1\right)}\)
Counting Sequence
1, 1, 2, 6, 19, 57, 165, 471, 1339, 3803, 10799, 30663, 87063, 247199, 701871, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x -1\right) \left(2 x^{3}-4 x^{2}+4 x -1\right) F \! \left(x \right)-x^{4}+2 x^{3}-5 x^{2}+4 x -1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(n +3\right) = 2 a \! \left(n \right)-4 a \! \left(n +1\right)+4 a \! \left(n +2\right)+1, \quad n \geq 5\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(n +3\right) = 2 a \! \left(n \right)-4 a \! \left(n +1\right)+4 a \! \left(n +2\right)+1, \quad n \geq 5\)
Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(33 \left(\left(\mathrm{I}-\frac{7 \sqrt{11}}{33}\right) \sqrt{3}-\frac{7 \,\mathrm{I} \sqrt{11}}{11}+1\right) 2^{\frac{2}{3}} \left(13+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}+1056-66 \left(\left(\mathrm{I}-\frac{\sqrt{11}}{33}\right) \sqrt{3}+\frac{\mathrm{I} \sqrt{11}}{11}-1\right) 2^{\frac{1}{3}} \left(13+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}\right) \left(\frac{13 \,2^{\frac{2}{3}} \left(\left(\mathrm{I}-\frac{3 \sqrt{11}}{13}\right) \sqrt{3}-\frac{9 \,\mathrm{I} \sqrt{11}}{13}+1\right) \left(13+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{384}-\frac{\mathrm{I} \sqrt{3}\, \left(26+6 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{\left(26+6 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{2}{3}\right)^{-n}}{2112}\\+\\\frac{\left(66 \left(\left(\mathrm{I}+\frac{\sqrt{11}}{33}\right) \sqrt{3}+\frac{\mathrm{I} \sqrt{11}}{11}+1\right) 2^{\frac{1}{3}} \left(13+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}+1056-33 \,2^{\frac{2}{3}} \left(\left(\mathrm{I}+\frac{7 \sqrt{11}}{33}\right) \sqrt{3}-\frac{7 \,\mathrm{I} \sqrt{11}}{11}-1\right) \left(13+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(-\frac{13 \,2^{\frac{2}{3}} \left(\left(\mathrm{I}+\frac{3 \sqrt{11}}{13}\right) \sqrt{3}-\frac{9 \,\mathrm{I} \sqrt{11}}{13}-1\right) \left(13+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{384}+\frac{\mathrm{I} \sqrt{3}\, \left(26+6 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{\left(26+6 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{2}{3}\right)^{-n}}{2112}\\-1+\\\frac{\left(\left(-4 \,2^{\frac{1}{3}} \sqrt{11}\, \sqrt{3}-132 \,2^{\frac{1}{3}}\right) \left(13+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}+1056+\left(14 \sqrt{11}\, \sqrt{3}\, 2^{\frac{2}{3}}-66 \,2^{\frac{2}{3}}\right) \left(13+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{2^{\frac{2}{3}} \left(3 \sqrt{11}\, \sqrt{3}-13\right) \left(13+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{192}-\frac{\left(26+6 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}+\frac{2}{3}\right)^{-n}}{2112} & \text{otherwise} \end{array}\right.\)
This specification was found using the strategy pack "Point Placements" and has 80 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_{12}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{12}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{12}\! \left(x \right) &= x\\
F_{13}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{15}\! \left(x \right) &= 0\\
F_{16}\! \left(x \right) &= F_{10}\! \left(x \right) F_{12}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{12}\! \left(x \right) F_{13}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{12}\! \left(x \right) F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{26}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{12}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{12}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{28}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{12}\! \left(x \right) F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{23}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{27}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{12}\! \left(x \right) F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{35}\! \left(x \right)+F_{39}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{12}\! \left(x \right) F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{12}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{34}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{12}\! \left(x \right) F_{19}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{41}\! \left(x \right)+F_{75}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{12}\! \left(x \right) F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{48}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{12}\! \left(x \right) F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{50}\! \left(x \right)+F_{54}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{12}\! \left(x \right) F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{44}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{49}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{12}\! \left(x \right) F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)+F_{69}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{57}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{12}\! \left(x \right) F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{59}\! \left(x \right)+F_{61}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{45}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{56}\! \left(x \right)+F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{54}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{12}\! \left(x \right) F_{64}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{65}\! \left(x \right)+F_{66}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{60}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{62}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{12}\! \left(x \right) F_{68}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{69}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{70}\! \left(x \right)+F_{74}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{12}\! \left(x \right) F_{71}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{72}\! \left(x \right)+F_{73}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{47}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{69}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{12}\! \left(x \right) F_{56}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{12}\! \left(x \right) F_{76}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{68}\! \left(x \right)+F_{77}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{78}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{70}\! \left(x \right)+F_{74}\! \left(x \right)+F_{79}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{12}\! \left(x \right) F_{77}\! \left(x \right)\\
\end{align*}\)