Av(1234, 1243, 1432, 2314, 4213)
Generating Function
\(\displaystyle \frac{2 x^{8}+5 x^{7}+10 x^{6}+11 x^{5}+6 x^{4}+x^{3}-x +1}{\left(x -1\right) \left(x^{2}+1\right) \left(x^{3}+2 x^{2}+x -1\right)}\)
Counting Sequence
1, 1, 2, 6, 19, 50, 114, 249, 542, 1173, 2526, 5430, 11670, 25075, 53865, ...
Implicit Equation for the Generating Function
\(\displaystyle -\left(x -1\right) \left(x^{2}+1\right) \left(x^{3}+2 x^{2}+x -1\right) F \! \left(x \right)+2 x^{8}+5 x^{7}+10 x^{6}+11 x^{5}+6 x^{4}+x^{3}-x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 50\)
\(\displaystyle a \! \left(6\right) = 114\)
\(\displaystyle a \! \left(7\right) = 249\)
\(\displaystyle a \! \left(8\right) = 542\)
\(\displaystyle a \! \left(n +5\right) = a \! \left(n \right)+2 a \! \left(n +1\right)+2 a \! \left(n +2\right)+a \! \left(n +3\right)+a \! \left(n +4\right)+35, \quad n \geq 9\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 50\)
\(\displaystyle a \! \left(6\right) = 114\)
\(\displaystyle a \! \left(7\right) = 249\)
\(\displaystyle a \! \left(8\right) = 542\)
\(\displaystyle a \! \left(n +5\right) = a \! \left(n \right)+2 a \! \left(n +1\right)+2 a \! \left(n +2\right)+a \! \left(n +3\right)+a \! \left(n +4\right)+35, \quad n \geq 9\)
Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0 \\ 1 & n =1 \\ 2 & n =2 \\ \frac{\left(2170 \,2^{\frac{1}{3}} \left(\left(\mathrm{I}+\frac{16 \sqrt{31}}{155}\right) \sqrt{3}-\frac{48 \,\mathrm{I} \sqrt{31}}{155}-1\right) \left(29+3 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}+496-217 \left(\left(\mathrm{I}-\frac{\sqrt{31}}{31}\right) \sqrt{3}-\frac{3 \,\mathrm{I} \sqrt{31}}{31}+1\right) 2^{\frac{2}{3}} \left(29+3 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}\right) \left(\frac{29 \left(\left(\mathrm{I}+\frac{3 \sqrt{31}}{29}\right) \sqrt{3}-\frac{9 \,\mathrm{I} \sqrt{31}}{29}-1\right) 2^{\frac{1}{3}} \left(29+3 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}}{24}-\frac{\mathrm{I} \sqrt{3}\, \left(116+12 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}-\frac{\left(116+12 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}-\frac{2}{3}\right)^{-n}}{2232}\\+\\\frac{\left(217 \left(\left(\mathrm{I}+\frac{\sqrt{31}}{31}\right) \sqrt{3}-\frac{3 \,\mathrm{I} \sqrt{31}}{31}-1\right) 2^{\frac{2}{3}} \left(29+3 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}+496-2170 \,2^{\frac{1}{3}} \left(\left(\mathrm{I}-\frac{16 \sqrt{31}}{155}\right) \sqrt{3}-\frac{48 \,\mathrm{I} \sqrt{31}}{155}+1\right) \left(29+3 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(-\frac{29 \left(\left(\mathrm{I}-\frac{3 \sqrt{31}}{29}\right) \sqrt{3}-\frac{9 \,\mathrm{I} \sqrt{31}}{29}+1\right) 2^{\frac{1}{3}} \left(29+3 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}}{24}+\frac{\mathrm{I} \sqrt{3}\, \left(116+12 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}-\frac{\left(116+12 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}-\frac{2}{3}\right)^{-n}}{2232}\\+\\\frac{\left(\left(-14 \sqrt{3}\, \sqrt{31}\, 2^{\frac{2}{3}}+434 \,2^{\frac{2}{3}}\right) \left(29+3 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}+496+\left(-448 \sqrt{31}\, \sqrt{3}\, 2^{\frac{1}{3}}+4340 \,2^{\frac{1}{3}}\right) \left(29+3 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{\left(-3 \sqrt{31}\, \sqrt{3}+29\right) 2^{\frac{1}{3}} \left(29+3 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}}{12}+\frac{\left(116+12 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{2}{3}\right)^{-n}}{2232}\\-\frac{5 \cos \left(\frac{n \pi}{2}\right)}{6}+\frac{\sin \left(\frac{n \pi}{2}\right)}{2}-\frac{35}{6} & \text{otherwise} \end{array}\right.\)
This specification was found using the strategy pack "Point Placements" and has 89 rules.
Found on January 18, 2022.Finding the specification took 2 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_{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_{14}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{10}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{18}\! \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_{4}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{12}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{41}\! \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_{29}\! \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_{26}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{20}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{39}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{33}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{37}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{42}\! \left(x \right)+F_{69}\! \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_{55}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{15}\! \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_{17}\! \left(x \right)+F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{14}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{4}\! \left(x \right) F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{27}\! \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_{16}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{4}\! \left(x \right) F_{52}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{57}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{4}\! \left(x \right) F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{59}\! \left(x \right)+F_{60}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{34}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{40}\! \left(x \right)+F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{4}\! \left(x \right) F_{59}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{4}\! \left(x \right) F_{64}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{66}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{62}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{68}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{4}\! \left(x \right) F_{66}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{4}\! \left(x \right) F_{70}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{71}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{72}\! \left(x \right)+F_{87}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{38}\! \left(x \right)+F_{73}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{4}\! \left(x \right) F_{74}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{75}\! \left(x \right)+F_{81}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{76}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{77}\! \left(x \right)+F_{80}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{4}\! \left(x \right) F_{78}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{79}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{28}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{27}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{82}\! \left(x \right)\\
F_{82}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{83}\! \left(x \right)+F_{86}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{4}\! \left(x \right) F_{84}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{59}\! \left(x \right)+F_{85}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{39}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{36}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{87}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{62}\! \left(x \right)+F_{88}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{4}\! \left(x \right) F_{72}\! \left(x \right)\\
\end{align*}\)