Av(1243, 2431, 3214)
Generating Function
\(\displaystyle \frac{x^{10}-2 x^{9}-x^{8}-13 x^{7}+54 x^{6}-99 x^{5}+108 x^{4}-77 x^{3}+35 x^{2}-9 x +1}{\left(x -1\right)^{2} \left(3 x^{3}-5 x^{2}+4 x -1\right)^{2}}\)
Counting Sequence
1, 1, 2, 6, 21, 72, 229, 683, 1954, 5452, 14974, 40671, 109509, 292743, 777810, ...
Implicit Equation for the Generating Function
\(\displaystyle -\left(x -1\right)^{2} \left(3 x^{3}-5 x^{2}+4 x -1\right)^{2} F \! \left(x \right)+x^{10}-2 x^{9}-x^{8}-13 x^{7}+54 x^{6}-99 x^{5}+108 x^{4}-77 x^{3}+35 x^{2}-9 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 72\)
\(\displaystyle a \! \left(6\right) = 229\)
\(\displaystyle a \! \left(7\right) = 683\)
\(\displaystyle a \! \left(8\right) = 1954\)
\(\displaystyle a \! \left(9\right) = 5452\)
\(\displaystyle a \! \left(10\right) = 14974\)
\(\displaystyle a \! \left(n +6\right) = -9 a \! \left(n \right)+30 a \! \left(n +1\right)-49 a \! \left(n +2\right)+46 a \! \left(n +3\right)-26 a \! \left(n +4\right)+8 a \! \left(n +5\right)-2 n +2, \quad n \geq 11\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 72\)
\(\displaystyle a \! \left(6\right) = 229\)
\(\displaystyle a \! \left(7\right) = 683\)
\(\displaystyle a \! \left(8\right) = 1954\)
\(\displaystyle a \! \left(9\right) = 5452\)
\(\displaystyle a \! \left(10\right) = 14974\)
\(\displaystyle a \! \left(n +6\right) = -9 a \! \left(n \right)+30 a \! \left(n +1\right)-49 a \! \left(n +2\right)+46 a \! \left(n +3\right)-26 a \! \left(n +4\right)+8 a \! \left(n +5\right)-2 n +2, \quad n \geq 11\)
Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0 \\ 1 & n =1 \\ 2 & n =2 \\ \frac{\left(-155558 \left(\left(\left(-\frac{102 n}{2509}+\frac{190089}{155558}\right) \sqrt{31}+\mathrm{I} n +\frac{11733 \,\mathrm{I}}{5018}\right) \sqrt{3}+\left(-\frac{306 \,\mathrm{I} n}{2509}+\frac{570267 \,\mathrm{I}}{155558}\right) \sqrt{31}+n +\frac{11733}{5018}\right) 2^{\frac{1}{3}} \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}-91729 \left(\left(\left(\frac{147 n}{269}+\frac{50445}{8339}\right) \sqrt{31}+\mathrm{I} n +\frac{23487 \,\mathrm{I}}{269}\right) \sqrt{3}+\left(-\frac{441 \,\mathrm{I} n}{269}-\frac{151335 \,\mathrm{I}}{8339}\right) \sqrt{31}-n -\frac{23487}{269}\right) 2^{\frac{2}{3}} \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}+28657640 n -106048272\right) \left(\frac{47 \,2^{\frac{1}{3}} \left(\left(\mathrm{I}-\frac{9 \sqrt{31}}{47}\right) \sqrt{3}-\frac{27 \,\mathrm{I} \sqrt{31}}{47}+1\right) \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}}{8712}-\frac{\mathrm{I} \sqrt{3}\, \left(188+36 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{36}+\frac{\left(188+36 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{36}+\frac{5}{9}\right)^{-n}}{226050264}\\+\\\frac{\left(91729 \left(\left(\left(-\frac{147 n}{269}-\frac{50445}{8339}\right) \sqrt{31}+\mathrm{I} n +\frac{23487 \,\mathrm{I}}{269}\right) \sqrt{3}+\left(-\frac{441 \,\mathrm{I} n}{269}-\frac{151335 \,\mathrm{I}}{8339}\right) \sqrt{31}+n +\frac{23487}{269}\right) 2^{\frac{2}{3}} \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}+155558 \left(\left(\left(\frac{102 n}{2509}-\frac{190089}{155558}\right) \sqrt{31}+\mathrm{I} n +\frac{11733 \,\mathrm{I}}{5018}\right) \sqrt{3}+\left(-\frac{306 \,\mathrm{I} n}{2509}+\frac{570267 \,\mathrm{I}}{155558}\right) \sqrt{31}-n -\frac{11733}{5018}\right) 2^{\frac{1}{3}} \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}+28657640 n -106048272\right) \left(-\frac{47 \left(\left(\mathrm{I}+\frac{9 \sqrt{31}}{47}\right) \sqrt{3}-\frac{27 \,\mathrm{I} \sqrt{31}}{47}-1\right) 2^{\frac{1}{3}} \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}}{8712}+\frac{\mathrm{I} \sqrt{3}\, \left(188+36 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{36}+\frac{\left(188+36 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{36}+\frac{5}{9}\right)^{-n}}{226050264}\\+\\\frac{\left(100254 \left(\sqrt{3}\, \left(n +\frac{16815}{1519}\right) \sqrt{31}-\frac{269 n}{147}-\frac{7829}{49}\right) 2^{\frac{2}{3}} \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}-12648 \left(\sqrt{3}\, \left(n -\frac{63363}{2108}\right) \sqrt{31}-\frac{2509 n}{102}-\frac{3911}{68}\right) 2^{\frac{1}{3}} \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}+28657640 n -106048272\right) \left(\frac{2^{\frac{1}{3}} \left(9 \sqrt{31}\, \sqrt{3}-47\right) \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}}{4356}-\frac{\left(188+36 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{18}+\frac{5}{9}\right)^{-n}}{226050264}\\-2 n +2 & \text{otherwise} \end{array}\right.\)
This specification was found using the strategy pack "Point Placements" and has 91 rules.
Found on January 18, 2022.Finding the specification took 2 seconds.
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Copy 91 equations to clipboard:
\(\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_{28}\! \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_{7}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{27}\! \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_{18}\! \left(x \right)+F_{21}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{18}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{25}\! \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_{22}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{21}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{13}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{30}\! \left(x \right)+F_{75}\! \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_{47}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{40}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{41}\! \left(x \right)+F_{46}\! \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_{19}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{4}\! \left(x \right) F_{43}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{36}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)+F_{54}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{4}\! \left(x \right) F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{4}\! \left(x \right) F_{51}\! \left(x \right)\\
F_{54}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{55}\! \left(x \right)+F_{70}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{4}\! \left(x \right) F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)+F_{60}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{52}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{4}\! \left(x \right) F_{57}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{61}\! \left(x \right)+F_{68}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{4}\! \left(x \right) F_{63}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{61}\! \left(x \right)+F_{64}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{65}\! \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_{67}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{69}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{4}\! \left(x \right) F_{60}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{4}\! \left(x \right) F_{71}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{72}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{64}\! \left(x \right)+F_{73}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{74}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{4}\! \left(x \right) F_{72}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{4}\! \left(x \right) F_{76}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{77}\! \left(x \right)+F_{85}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{78}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{79}\! \left(x \right)+F_{84}\! \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_{11}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{52}\! \left(x \right)+F_{83}\! \left(x \right)\\
F_{83}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{55}\! \left(x \right)+F_{74}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{4}\! \left(x \right) F_{77}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{64}\! \left(x \right)+F_{86}\! \left(x \right)\\
F_{86}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{87}\! \left(x \right)+F_{90}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{4}\! \left(x \right) F_{88}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{89}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{4}\! \left(x \right) F_{85}\! \left(x \right)\\
\end{align*}\)