Av(1234, 1342, 3142, 3214)
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Generating Function
\(\displaystyle -\frac{\left(x -1\right)^{3}}{\left(x^{3}-3 x^{2}+2 x -1\right) \left(x^{3}+2 x^{2}+2 x -1\right)}\)
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Counting Sequence
1, 1, 2, 6, 20, 61, 172, 476, 1332, 3772, 10717, 30399, 86067, 243541, 689299, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(x^{3}-3 x^{2}+2 x -1\right) \left(x^{3}+2 x^{2}+2 x -1\right) F \! \left(x \right)+\left(x -1\right)^{3} = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 61\)
\(\displaystyle a \! \left(n +6\right) = -a \! \left(n \right)+a \! \left(n +1\right)+2 a \! \left(n +2\right)+4 a \! \left(n +3\right)-5 a \! \left(n +4\right)+4 a \! \left(n +5\right), \quad n \geq 6\)
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Explicit Closed Form
\(\displaystyle -\frac{114 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +4}\right)}{9545}+\frac{4 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +3}\right)}{47725}+\frac{131 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +2}\right)}{9545}+\frac{2562 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +1}\right)}{47725}+\frac{1253 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n}\right)}{9545}+\frac{2908 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{6}-Z^{5}-2 Z^{4}-4 Z^{3}+5 Z^{2}-4 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n -1}\right)}{47725}\)
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This specification was found using the strategy pack "Point Placements" and has 54 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_{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_{15}\! \left(x \right) &= F_{16}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{11}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{20}\! \left(x \right)+F_{38}\! \left(x \right)\\ F_{19}\! \left(x \right) &= 0\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{24}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{35}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{31}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{28}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{34}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{29}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{37}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{29}\! \left(x \right)\\ 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_{48}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{42}\! \left(x \right)+F_{47}\! \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_{45}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{14}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{46}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{36}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{4}\! \left(x \right) F_{40}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{49}\! \left(x \right)\\ F_{49}\! \left(x \right) &= 2 F_{19}\! \left(x \right)+F_{36}\! \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_{52}\! \left(x \right)+F_{53}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{14}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{32}\! \left(x \right)\\ \end{align*}\)