Av(1234, 1243, 2341, 2413)
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Generating Function
\(\displaystyle \frac{x^{8}-7 x^{7}+31 x^{6}-74 x^{5}+106 x^{4}-88 x^{3}+41 x^{2}-10 x +1}{\left(2 x -1\right) \left(x^{2}-3 x +1\right)^{2} \left(-1+x \right)^{3}}\)
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Counting Sequence
1, 1, 2, 6, 20, 64, 194, 565, 1603, 4473, 12349, 33853, 92351, 251044, 680606, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(2 x -1\right) \left(x^{2}-3 x +1\right)^{2} \left(-1+x \right)^{3} F \! \left(x \right)-x^{8}+7 x^{7}-31 x^{6}+74 x^{5}-106 x^{4}+88 x^{3}-41 x^{2}+10 x -1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 64\)
\(\displaystyle a \! \left(6\right) = 194\)
\(\displaystyle a \! \left(7\right) = 565\)
\(\displaystyle a \! \left(8\right) = 1603\)
\(\displaystyle a \! \left(n +5\right) = \frac{n^{2}}{2}+2 a \! \left(n \right)-13 a \! \left(n +1\right)+28 a \! \left(n +2\right)-23 a \! \left(n +3\right)+8 a \! \left(n +4\right)-\frac{n}{2}-2, \quad n \geq 9\)
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Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(\left(-15 n -21\right) \sqrt{5}+35 n +75\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{50}+\\\frac{\left(\left(15 n +21\right) \sqrt{5}+35 n +75\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{50}-\frac{n^{2}}{2}+\frac{3 n}{2}-2^{-1+n}-2 & \text{otherwise} \end{array}\right.\)
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This specification was found using the strategy pack "Point Placements" and has 84 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_{10}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{19}\! \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_{11}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x \right)\\ F_{10}\! \left(x \right) &= x\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{14}\! \left(x \right)+F_{18}\! \left(x \right)\\ F_{13}\! \left(x \right) &= 0\\ F_{14}\! \left(x \right) &= F_{10}\! \left(x \right) F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{17}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{10}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{12}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{10}\! \left(x \right) F_{6}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{33}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{10}\! \left(x \right) F_{22}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{26}\! \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_{10}\! \left(x \right) F_{23}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{28}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{10}\! \left(x \right) F_{29}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{31}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{24}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{27}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{10}\! \left(x \right) F_{26}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{34}\! \left(x \right)+F_{58}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{10}\! \left(x \right) F_{35}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{41}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{10}\! \left(x \right) F_{39}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{38}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= 2 F_{13}\! \left(x \right)+F_{43}\! \left(x \right)+F_{47}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{10}\! \left(x \right) F_{44}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{46}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{37}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{42}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{10}\! \left(x \right) F_{48}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)+F_{53}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{50}\! \left(x \right)+F_{52}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{10}\! \left(x \right) F_{51}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{48}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{10}\! \left(x \right) F_{20}\! \left(x \right)\\ F_{53}\! \left(x \right) &= 2 F_{13}\! \left(x \right)+F_{47}\! \left(x \right)+F_{54}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{10}\! \left(x \right) F_{55}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)+F_{57}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{40}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{53}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{10}\! \left(x \right) F_{59}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{60}\! \left(x \right)+F_{61}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{49}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{62}\! \left(x \right)+F_{79}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{28}\! \left(x \right)+F_{63}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{10}\! \left(x \right) F_{64}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{65}\! \left(x \right)+F_{71}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{66}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{52}\! \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_{69}\! \left(x \right)+F_{70}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{10}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{66}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{62}\! \left(x \right)+F_{72}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{73}\! \left(x \right)+F_{74}\! \left(x \right)+F_{78}\! \left(x \right)\\ F_{73}\! \left(x \right) &= 0\\ 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_{77}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{66}\! \left(x \right)\\ F_{77}\! \left(x \right) &= F_{72}\! \left(x \right)\\ F_{78}\! \left(x \right) &= F_{10}\! \left(x \right) F_{62}\! \left(x \right)\\ F_{79}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{54}\! \left(x \right)+F_{78}\! \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_{82}\! \left(x \right)+F_{83}\! \left(x \right)\\ F_{82}\! \left(x \right) &= F_{49}\! \left(x \right)\\ F_{83}\! \left(x \right) &= F_{79}\! \left(x \right)\\ \end{align*}\)