Av(1342, 1432, 3241, 3412)
Generating Function
\(\displaystyle \frac{x^{6}-3 x^{5}+5 x^{4}-7 x^{3}+9 x^{2}-5 x +1}{\left(x^{3}-x^{2}+3 x -1\right) \left(x -1\right)^{3}}\)
Counting Sequence
1, 1, 2, 6, 20, 62, 182, 519, 1458, 4065, 11292, 31314, 86770, 240354, 665684, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{3}-x^{2}+3 x -1\right) \left(x -1\right)^{3} F \! \left(x \right)-x^{6}+3 x^{5}-5 x^{4}+7 x^{3}-9 x^{2}+5 x -1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 62\)
\(\displaystyle a \! \left(6\right) = 182\)
\(\displaystyle a \! \left(n +3\right) = a \! \left(n \right)-a \! \left(n +1\right)+3 a \! \left(n +2\right)+\frac{\left(n +1\right) \left(n +2\right)}{2}, \quad n \geq 7\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 62\)
\(\displaystyle a \! \left(6\right) = 182\)
\(\displaystyle a \! \left(n +3\right) = a \! \left(n \right)-a \! \left(n +1\right)+3 a \! \left(n +2\right)+\frac{\left(n +1\right) \left(n +2\right)}{2}, \quad n \geq 7\)
Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0 \\ \frac{7 \left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{1}{3}} \left(\left(\left(\mathrm{I}+\frac{15 \sqrt{19}}{133}\right) \sqrt{3}-\frac{45 \,\mathrm{I} \sqrt{19}}{133}-1\right) \left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{1}{3}}+\left(-\frac{16 \,\mathrm{I}}{7}-\frac{48 \sqrt{19}}{133}\right) \sqrt{3}-\frac{144 \,\mathrm{I} \sqrt{19}}{133}-\frac{16}{7}\right) \left(\frac{\left(\left(\mathrm{I}+3 \sqrt{19}\right) \sqrt{3}-9 \,\mathrm{I} \sqrt{19}-1\right) \left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{2}{3}}}{384}-\frac{\mathrm{I} \sqrt{3}\, \left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{\left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}+\frac{1}{3}\right)^{-n}}{512}\\-\\\frac{7 \left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{1}{3}} \left(\left(\left(\mathrm{I}-\frac{15 \sqrt{19}}{133}\right) \sqrt{3}-\frac{45 \,\mathrm{I} \sqrt{19}}{133}+1\right) \left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{1}{3}}+\left(-\frac{16 \,\mathrm{I}}{7}+\frac{48 \sqrt{19}}{133}\right) \sqrt{3}-\frac{144 \,\mathrm{I} \sqrt{19}}{133}+\frac{16}{7}\right) \left(\frac{\left(\left(-\mathrm{I}+3 \sqrt{19}\right) \sqrt{3}+9 \,\mathrm{I} \sqrt{19}-1\right) \left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{2}{3}}}{384}+\frac{\mathrm{I} \sqrt{3}\, \left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{\left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}+\frac{1}{3}\right)^{-n}}{512}\\+\\\frac{\left(96 \left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{1}{3}} \sqrt{19}\, \sqrt{3}-30 \left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{19}\, \sqrt{3}+608 \left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{1}{3}}+266 \left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{\left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{1}{3}}}{3}+\frac{1}{3}+\frac{\left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{2}{3}}}{192}-\frac{\left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{19}\, \sqrt{3}}{64}\right)^{-n}}{9728}\\-\frac{n^{2}}{4}-\frac{n}{4} & \text{otherwise} \end{array}\right.\)
This specification was found using the strategy pack "Point Placements" and has 82 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_{10}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{16}\! \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_{10}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{5}\! \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) &= x\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{13}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{12}\! \left(x \right) &= 0\\
F_{13}\! \left(x \right) &= F_{10}\! \left(x \right) F_{6}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{10}\! \left(x \right) F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{9}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{10}\! \left(x \right) F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{10}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{10}\! \left(x \right) F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{24}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{10}\! \left(x \right) F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{10}\! \left(x \right) F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{33}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{30}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{10}\! \left(x \right) F_{33}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{10}\! \left(x \right) F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{37}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{44}\! \left(x \right)+F_{62}\! \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_{54}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{14}\! \left(x \right)+F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{10}\! \left(x \right) F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{10}\! \left(x \right) F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{50}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= 2 F_{12}\! \left(x \right)+F_{56}\! \left(x \right)+F_{74}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{10}\! \left(x \right) F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)+F_{78}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{59}\! \left(x \right)+F_{77}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{60}\! \left(x \right)+F_{62}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{10}\! \left(x \right) F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{59}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{10}\! \left(x \right) F_{63}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{64}\! \left(x \right)+F_{65}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{66}\! \left(x \right)+F_{69}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{39}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{10}\! \left(x \right) F_{68}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{70}\! \left(x \right)+F_{73}\! \left(x \right)+F_{74}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{10}\! \left(x \right) F_{71}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{72}\! \left(x \right)\\
F_{72}\! \left(x \right) &= 2 F_{12}\! \left(x \right)+F_{73}\! \left(x \right)+F_{74}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{10}\! \left(x \right) F_{59}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{10}\! \left(x \right) F_{75}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{76}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{72}\! \left(x \right)\\
F_{77}\! \left(x \right) &= 2 F_{12}\! \left(x \right)+F_{56}\! \left(x \right)+F_{74}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{72}\! \left(x \right)+F_{79}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{80}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{10}\! \left(x \right) F_{81}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{78}\! \left(x \right)\\
\end{align*}\)