Av(1234, 1342, 4213)
Generating Function
\(\displaystyle \frac{x^{10}-6 x^{9}+9 x^{8}+9 x^{7}-54 x^{6}+94 x^{5}-104 x^{4}+76 x^{3}-35 x^{2}+9 x -1}{\left(2 x -1\right) \left(x^{3}-2 x^{2}+3 x -1\right) \left(-1+x \right)^{5}}\)
Counting Sequence
1, 1, 2, 6, 21, 72, 228, 678, 1929, 5307, 14203, 37133, 95179, 239942, 596587, ...
Implicit Equation for the Generating Function
\(\displaystyle -\left(2 x -1\right) \left(x^{3}-2 x^{2}+3 x -1\right) \left(-1+x \right)^{5} F \! \left(x \right)+x^{10}-6 x^{9}+9 x^{8}+9 x^{7}-54 x^{6}+94 x^{5}-104 x^{4}+76 x^{3}-35 x^{2}+9 x -1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 72\)
\(\displaystyle a \! \left(6\right) = 228\)
\(\displaystyle a \! \left(7\right) = 678\)
\(\displaystyle a \! \left(8\right) = 1929\)
\(\displaystyle a \! \left(9\right) = 5307\)
\(\displaystyle a \! \left(10\right) = 14203\)
\(\displaystyle a \! \left(n +4\right) = \frac{n^{4}}{12}+\frac{n^{3}}{6}+\frac{5 n^{2}}{12}-2 a \! \left(n \right)+5 a \! \left(n +1\right)-8 a \! \left(n +2\right)+5 a \! \left(n +3\right)+\frac{n}{3}+5, \quad n \geq 11\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 72\)
\(\displaystyle a \! \left(6\right) = 228\)
\(\displaystyle a \! \left(7\right) = 678\)
\(\displaystyle a \! \left(8\right) = 1929\)
\(\displaystyle a \! \left(9\right) = 5307\)
\(\displaystyle a \! \left(10\right) = 14203\)
\(\displaystyle a \! \left(n +4\right) = \frac{n^{4}}{12}+\frac{n^{3}}{6}+\frac{5 n^{2}}{12}-2 a \! \left(n \right)+5 a \! \left(n +1\right)-8 a \! \left(n +2\right)+5 a \! \left(n +3\right)+\frac{n}{3}+5, \quad n \geq 11\)
Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0 \\ 1 & n =1 \\ \frac{\left(-1909 \left(\left(\mathrm{I}-\frac{107 \sqrt{23}}{1909}\right) \sqrt{3}-\frac{321 \,\mathrm{I} \sqrt{23}}{1909}+1\right) 2^{\frac{1}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}+18400+115 \left(\left(\mathrm{I}-\frac{91 \sqrt{23}}{23}\right) \sqrt{3}+\frac{273 \,\mathrm{I} \sqrt{23}}{23}-1\right) 2^{\frac{2}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{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(-115 \left(\left(\mathrm{I}+\frac{91 \sqrt{23}}{23}\right) \sqrt{3}+\frac{273 \,\mathrm{I} \sqrt{23}}{23}+1\right) 2^{\frac{2}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}+18400+1909 \,2^{\frac{1}{3}} \left(\left(\mathrm{I}+\frac{107 \sqrt{23}}{1909}\right) \sqrt{3}-\frac{321 \,\mathrm{I} \sqrt{23}}{1909}-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(910 \,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}}+18400+\left(-214 \sqrt{23}\, \sqrt{3}\, 2^{\frac{1}{3}}+3818 \,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^{4}}{12}+\frac{n^{3}}{6}+\frac{17 n^{2}}{12}+\frac{16 n}{3}-\frac{63 \,2^{n}}{4}+12 & \text{otherwise} \end{array}\right.\)
This specification was found using the strategy pack "Point Placements" and has 147 rules.
Found on January 18, 2022.Finding the specification took 11 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_{23}\! \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_{13}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{10}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{18}\! \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_{10}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{14}\! \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_{22}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{36}\! \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_{27}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{32}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{29}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{118}\! \left(x \right)+F_{37}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{37}\! \left(x \right) &= 0\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{62}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{42}\! \left(x \right)+F_{49}\! \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_{48}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{4}\! \left(x \right) F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{4}\! \left(x \right) F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{53}\! \left(x \right)+F_{55}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{4}\! \left(x \right) F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{44}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{4}\! \left(x \right) F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{52}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{4}\! \left(x \right) F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{58}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{63}\! \left(x \right) &= 2 F_{37}\! \left(x \right)+F_{106}\! \left(x \right)+F_{64}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{4}\! \left(x \right) F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{66}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{67}\! \left(x \right)+F_{96}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{68}\! \left(x \right)+F_{86}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{4}\! \left(x \right) F_{69}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{70}\! \left(x \right)+F_{75}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{71}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{42}\! \left(x \right)+F_{72}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{4}\! \left(x \right) F_{73}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{74}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{53}\! \left(x \right)+F_{72}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{67}\! \left(x \right)+F_{76}\! \left(x \right)\\
F_{76}\! \left(x \right) &= 2 F_{37}\! \left(x \right)+F_{64}\! \left(x \right)+F_{77}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{4}\! \left(x \right) F_{78}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{79}\! \left(x \right)\\
F_{79}\! \left(x \right) &= 2 F_{37}\! \left(x \right)+F_{77}\! \left(x \right)+F_{80}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{4}\! \left(x \right) F_{81}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{82}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{83}\! \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_{33}\! \left(x \right)+F_{83}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{4}\! \left(x \right) F_{87}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{88}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{86}\! \left(x \right)+F_{89}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{4}\! \left(x \right) F_{90}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{91}\! \left(x \right)+F_{92}\! \left(x \right)\\
F_{91}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{74}\! \left(x \right)\\
F_{92}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{93}\! \left(x \right)\\
F_{93}\! \left(x \right) &= 2 F_{37}\! \left(x \right)+F_{80}\! \left(x \right)+F_{94}\! \left(x \right)\\
F_{94}\! \left(x \right) &= F_{4}\! \left(x \right) F_{95}\! \left(x \right)\\
F_{95}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{93}\! \left(x \right)\\
F_{96}\! \left(x \right) &= F_{97}\! \left(x \right)\\
F_{97}\! \left(x \right) &= F_{4}\! \left(x \right) F_{98}\! \left(x \right)\\
F_{98}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{99}\! \left(x \right)\\
F_{99}\! \left(x \right) &= F_{97}\! \left(x \right)\\
F_{100}\! \left(x \right) &= F_{101}\! \left(x \right)\\
F_{101}\! \left(x \right) &= F_{102}\! \left(x \right)\\
F_{102}\! \left(x \right) &= F_{103}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{103}\! \left(x \right) &= F_{104}\! \left(x \right)+F_{105}\! \left(x \right)\\
F_{104}\! \left(x \right) &= F_{67}\! \left(x \right)\\
F_{105}\! \left(x \right) &= F_{101}\! \left(x \right)\\
F_{106}\! \left(x \right) &= F_{107}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{107}\! \left(x \right) &= F_{108}\! \left(x \right)+F_{109}\! \left(x \right)\\
F_{108}\! \left(x \right) &= F_{101}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{109}\! \left(x \right) &= F_{110}\! \left(x \right)+F_{114}\! \left(x \right)\\
F_{110}\! \left(x \right) &= 2 F_{37}\! \left(x \right)+F_{111}\! \left(x \right)+F_{80}\! \left(x \right)\\
F_{111}\! \left(x \right) &= F_{112}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{112}\! \left(x \right) &= F_{113}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{113}\! \left(x \right) &= F_{110}\! \left(x \right)\\
F_{114}\! \left(x \right) &= F_{115}\! \left(x \right)\\
F_{115}\! \left(x \right) &= F_{116}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{116}\! \left(x \right) &= F_{105}\! \left(x \right)+F_{117}\! \left(x \right)\\
F_{117}\! \left(x \right) &= F_{114}\! \left(x \right)\\
F_{118}\! \left(x \right) &= F_{119}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{119}\! \left(x \right) &= F_{120}\! \left(x \right)+F_{121}\! \left(x \right)\\
F_{120}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{121}\! \left(x \right) &= F_{122}\! \left(x \right)+F_{134}\! \left(x \right)\\
F_{122}\! \left(x \right) &= F_{123}\! \left(x \right)+F_{131}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{123}\! \left(x \right) &= F_{124}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{124}\! \left(x \right) &= F_{125}\! \left(x \right)+F_{126}\! \left(x \right)\\
F_{125}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{126}\! \left(x \right) &= F_{127}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{127}\! \left(x \right) &= 2 F_{37}\! \left(x \right)+F_{128}\! \left(x \right)+F_{80}\! \left(x \right)\\
F_{128}\! \left(x \right) &= F_{129}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{129}\! \left(x \right) &= F_{130}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{130}\! \left(x \right) &= F_{127}\! \left(x \right)\\
F_{131}\! \left(x \right) &= F_{132}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{132}\! \left(x \right) &= F_{133}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{133}\! \left(x \right) &= F_{122}\! \left(x \right)\\
F_{134}\! \left(x \right) &= 2 F_{37}\! \left(x \right)+F_{135}\! \left(x \right)+F_{144}\! \left(x \right)\\
F_{135}\! \left(x \right) &= F_{136}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{136}\! \left(x \right) &= F_{137}\! \left(x \right)+F_{139}\! \left(x \right)\\
F_{137}\! \left(x \right) &= F_{138}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{138}\! \left(x \right) &= F_{59}\! \left(x \right)\\
F_{139}\! \left(x \right) &= F_{101}\! \left(x \right)+F_{140}\! \left(x \right)\\
F_{140}\! \left(x \right) &= F_{141}\! \left(x \right)\\
F_{141}\! \left(x \right) &= F_{142}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{142}\! \left(x \right) &= F_{105}\! \left(x \right)+F_{143}\! \left(x \right)\\
F_{143}\! \left(x \right) &= F_{141}\! \left(x \right)\\
F_{144}\! \left(x \right) &= F_{145}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{145}\! \left(x \right) &= F_{104}\! \left(x \right)+F_{146}\! \left(x \right)\\
F_{146}\! \left(x \right) &= F_{134}\! \left(x \right)\\
\end{align*}\)