Av(1342, 1423, 3241, 4213)
Generating Function
\(\displaystyle \frac{x^{8}-6 x^{7}+19 x^{6}-40 x^{5}+54 x^{4}-49 x^{3}+27 x^{2}-8 x +1}{\left(2 x -1\right) \left(x^{3}-2 x^{2}+3 x -1\right) \left(x -1\right)^{4}}\)
Counting Sequence
1, 1, 2, 6, 20, 62, 176, 467, 1183, 2907, 7006, 16680, 39411, 92677, 217277, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(2 x -1\right) \left(x^{3}-2 x^{2}+3 x -1\right) \left(x -1\right)^{4} F \! \left(x \right)-x^{8}+6 x^{7}-19 x^{6}+40 x^{5}-54 x^{4}+49 x^{3}-27 x^{2}+8 x -1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 62\)
\(\displaystyle a \! \left(6\right) = 176\)
\(\displaystyle a \! \left(7\right) = 467\)
\(\displaystyle a \! \left(8\right) = 1183\)
\(\displaystyle a \! \left(n +4\right) = -2 a \! \left(n \right)+5 a \! \left(n +1\right)-8 a \! \left(n +2\right)+5 a \! \left(n +3\right)-\frac{\left(n -2\right) \left(n +3\right) \left(n +2\right)}{6}, \quad n \geq 9\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 62\)
\(\displaystyle a \! \left(6\right) = 176\)
\(\displaystyle a \! \left(7\right) = 467\)
\(\displaystyle a \! \left(8\right) = 1183\)
\(\displaystyle a \! \left(n +4\right) = -2 a \! \left(n \right)+5 a \! \left(n +1\right)-8 a \! \left(n +2\right)+5 a \! \left(n +3\right)-\frac{\left(n -2\right) \left(n +3\right) \left(n +2\right)}{6}, \quad n \geq 9\)
Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(-368 \,2^{\frac{1}{3}} \left(\left(-\frac{7 \sqrt{23}}{184}+\mathrm{I}\right) \sqrt{3}-\frac{21 \,\mathrm{I} \sqrt{23}}{184}+1\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}+9200-115 \,2^{\frac{2}{3}} \left(\left(\frac{19 \sqrt{23}}{23}+\mathrm{I}\right) \sqrt{3}-\frac{57 \,\mathrm{I} \sqrt{23}}{23}-1\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}\right) \left(\frac{11 \,2^{\frac{1}{3}} \left(\left(\mathrm{I}-\frac{3 \sqrt{23}}{11}\right) \sqrt{3}-\frac{9 \,\mathrm{I} \sqrt{23}}{11}+1\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}}{600}-\frac{\mathrm{I} \sqrt{3}\, \left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{\left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{2}{3}\right)^{-n}}{13800}\\+\\\frac{\left(115 \,2^{\frac{2}{3}} \left(\left(-\frac{19 \sqrt{23}}{23}+\mathrm{I}\right) \sqrt{3}-\frac{57 \,\mathrm{I} \sqrt{23}}{23}+1\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}+9200+368 \,2^{\frac{1}{3}} \left(\left(\frac{7 \sqrt{23}}{184}+\mathrm{I}\right) \sqrt{3}-\frac{21 \,\mathrm{I} \sqrt{23}}{184}-1\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(-\frac{11 \left(\left(\mathrm{I}+\frac{3 \sqrt{23}}{11}\right) \sqrt{3}-\frac{9 \,\mathrm{I} \sqrt{23}}{11}-1\right) 2^{\frac{1}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}}{600}+\frac{\mathrm{I} \sqrt{3}\, \left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{\left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{2}{3}\right)^{-n}}{13800}\\+\\\frac{\left(\left(190 \,2^{\frac{2}{3}} \sqrt{23}\, \sqrt{3}-230 \,2^{\frac{2}{3}}\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}+9200+\left(-28 \,2^{\frac{1}{3}} \sqrt{3}\, \sqrt{23}+736 \,2^{\frac{1}{3}}\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{2^{\frac{1}{3}} \left(3 \sqrt{23}\, \sqrt{3}-11\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}}{300}-\frac{\left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}+\frac{2}{3}\right)^{-n}}{13800}\\-\frac{n^{3}}{6}-\frac{n^{2}}{2}-\frac{n}{3}-2^{n -1}-1 & \text{otherwise} \end{array}\right.\)
This specification was found using the strategy pack "Point Placements" and has 101 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_{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_{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_{11}\! \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_{14}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \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_{26}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{22}\! \left(x \right)\\
F_{22}\! \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_{21}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{26}\! \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_{32}\! \left(x \right) &= F_{33}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{36}\! \left(x \right)+F_{73}\! \left(x \right)\\
F_{35}\! \left(x \right) &= 0\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{40}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{4}\! \left(x \right) F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{14}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{4}\! \left(x \right) F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{10}\! \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_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{4}\! \left(x \right) F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{4}\! \left(x \right) F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)+F_{69}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{4}\! \left(x \right) F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)+F_{61}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{4}\! \left(x \right) F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{56}\! \left(x \right)+F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{14}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{53}\! \left(x \right)+F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{63}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{4}\! \left(x \right) F_{64}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{61}\! \left(x \right)+F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{66}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{67}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{4}\! \left(x \right) F_{68}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{53}\! \left(x \right)+F_{66}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{70}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{4}\! \left(x \right) F_{71}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{52}\! \left(x \right)+F_{72}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{66}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{4}\! \left(x \right) F_{74}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{75}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{76}\! \left(x \right)+F_{85}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{77}\! \left(x \right)+F_{83}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{4}\! \left(x \right) F_{78}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{79}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{80}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{47}\! \left(x \right)+F_{81}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{4}\! \left(x \right) F_{82}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{79}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{4}\! \left(x \right) F_{84}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{76}\! \left(x \right)\\
F_{85}\! \left(x \right) &= 2 F_{35}\! \left(x \right)+F_{86}\! \left(x \right)+F_{92}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{4}\! \left(x \right) F_{87}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{88}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{89}\! \left(x \right)\\
F_{89}\! \left(x \right) &= 2 F_{35}\! \left(x \right)+F_{50}\! \left(x \right)+F_{90}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{4}\! \left(x \right) F_{91}\! \left(x \right)\\
F_{91}\! \left(x \right) &= F_{88}\! \left(x \right)\\
F_{92}\! \left(x \right) &= F_{4}\! \left(x \right) F_{93}\! \left(x \right)\\
F_{93}\! \left(x \right) &= F_{53}\! \left(x \right)+F_{94}\! \left(x \right)\\
F_{94}\! \left(x \right) &= 2 F_{35}\! \left(x \right)+F_{92}\! \left(x \right)+F_{95}\! \left(x \right)\\
F_{95}\! \left(x \right) &= F_{4}\! \left(x \right) F_{96}\! \left(x \right)\\
F_{96}\! \left(x \right) &= F_{97}\! \left(x \right)\\
F_{97}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{98}\! \left(x \right)\\
F_{98}\! \left(x \right) &= 2 F_{35}\! \left(x \right)+F_{50}\! \left(x \right)+F_{99}\! \left(x \right)\\
F_{99}\! \left(x \right) &= F_{100}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{100}\! \left(x \right) &= F_{97}\! \left(x \right)\\
\end{align*}\)