Av(1234, 1243, 1342, 3214)
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Generating Function
\(\displaystyle \frac{x^{4}-x^{3}-2 x +1}{2 x^{6}-3 x^{5}-x^{4}-2 x^{3}+x^{2}-3 x +1}\)
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Counting Sequence
1, 1, 2, 6, 20, 62, 181, 531, 1570, 4651, 13772, 40755, 120596, 356876, 1056127, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(2 x^{6}-3 x^{5}-x^{4}-2 x^{3}+x^{2}-3 x +1\right) F \! \left(x \right)-x^{4}+x^{3}+2 x -1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 62\)
\(\displaystyle a \! \left(n +6\right) = -2 a \! \left(n \right)+3 a \! \left(n +1\right)+a \! \left(n +2\right)+2 a \! \left(n +3\right)-a \! \left(n +4\right)+3 a \! \left(n +5\right), \quad n \geq 6\)
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Explicit Closed Form
\(\displaystyle \frac{858708 \mathit{RootOf} \left(2 Z^{6}-3 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =1\right)^{-n +4}}{150705721}+\frac{858708 \mathit{RootOf} \left(2 Z^{6}-3 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)^{-n +4}}{150705721}+\frac{858708 \mathit{RootOf} \left(2 Z^{6}-3 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)^{-n +4}}{150705721}+\frac{858708 \mathit{RootOf} \left(2 Z^{6}-3 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)^{-n +4}}{150705721}+\frac{858708 \mathit{RootOf} \left(2 Z^{6}-3 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =5\right)^{-n +4}}{150705721}+\frac{858708 \mathit{RootOf} \left(2 Z^{6}-3 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =6\right)^{-n +4}}{150705721}-\frac{11085262 \mathit{RootOf} \left(2 Z^{6}-3 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =1\right)^{-n +3}}{150705721}-\frac{11085262 \mathit{RootOf} \left(2 Z^{6}-3 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)^{-n +3}}{150705721}-\frac{11085262 \mathit{RootOf} \left(2 Z^{6}-3 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)^{-n +3}}{150705721}-\frac{11085262 \mathit{RootOf} \left(2 Z^{6}-3 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)^{-n +3}}{150705721}-\frac{11085262 \mathit{RootOf} \left(2 Z^{6}-3 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =5\right)^{-n +3}}{150705721}-\frac{11085262 \mathit{RootOf} \left(2 Z^{6}-3 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =6\right)^{-n +3}}{150705721}+\frac{3896392 \mathit{RootOf} \left(2 Z^{6}-3 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =1\right)^{-n +2}}{150705721}+\frac{3896392 \mathit{RootOf} \left(2 Z^{6}-3 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)^{-n +2}}{150705721}+\frac{3896392 \mathit{RootOf} \left(2 Z^{6}-3 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)^{-n +2}}{150705721}+\frac{3896392 \mathit{RootOf} \left(2 Z^{6}-3 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)^{-n +2}}{150705721}+\frac{3896392 \mathit{RootOf} \left(2 Z^{6}-3 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =5\right)^{-n +2}}{150705721}+\frac{3896392 \mathit{RootOf} \left(2 Z^{6}-3 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =6\right)^{-n +2}}{150705721}+\frac{377073 \mathit{RootOf} \left(2 Z^{6}-3 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =1\right)^{-n +1}}{5196749}+\frac{377073 \mathit{RootOf} \left(2 Z^{6}-3 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)^{-n +1}}{5196749}+\frac{377073 \mathit{RootOf} \left(2 Z^{6}-3 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)^{-n +1}}{5196749}+\frac{377073 \mathit{RootOf} \left(2 Z^{6}-3 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)^{-n +1}}{5196749}+\frac{377073 \mathit{RootOf} \left(2 Z^{6}-3 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =5\right)^{-n +1}}{5196749}+\frac{377073 \mathit{RootOf} \left(2 Z^{6}-3 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =6\right)^{-n +1}}{5196749}+\frac{960707 \mathit{RootOf} \left(2 Z^{6}-3 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =1\right)^{-n -1}}{150705721}+\frac{960707 \mathit{RootOf} \left(2 Z^{6}-3 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)^{-n -1}}{150705721}+\frac{960707 \mathit{RootOf} \left(2 Z^{6}-3 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)^{-n -1}}{150705721}+\frac{960707 \mathit{RootOf} \left(2 Z^{6}-3 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)^{-n -1}}{150705721}+\frac{960707 \mathit{RootOf} \left(2 Z^{6}-3 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =5\right)^{-n -1}}{150705721}+\frac{960707 \mathit{RootOf} \left(2 Z^{6}-3 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =6\right)^{-n -1}}{150705721}+\frac{33715409 \mathit{RootOf} \left(2 Z^{6}-3 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =1\right)^{-n}}{150705721}+\frac{33715409 \mathit{RootOf} \left(2 Z^{6}-3 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)^{-n}}{150705721}+\frac{33715409 \mathit{RootOf} \left(2 Z^{6}-3 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)^{-n}}{150705721}+\frac{33715409 \mathit{RootOf} \left(2 Z^{6}-3 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)^{-n}}{150705721}+\frac{33715409 \mathit{RootOf} \left(2 Z^{6}-3 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =5\right)^{-n}}{150705721}+\frac{33715409 \mathit{RootOf} \left(2 Z^{6}-3 Z^{5}-Z^{4}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =6\right)^{-n}}{150705721}\)
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This specification was found using the strategy pack "Point Placements" and has 62 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_{4}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{4}\! \left(x \right) &= x\\ F_{5}\! \left(x \right) &= F_{15}\! \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_{11}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{4}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{18}\! \left(x \right)+F_{44}\! \left(x \right)\\ F_{17}\! \left(x \right) &= 0\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{25}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{17}\! \left(x \right)+F_{22}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{24}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{21}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{36}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{2}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{33}\! \left(x \right)\\ F_{33}\! \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_{30}\! \left(x \right)\\ F_{36}\! \left(x \right) &= 2 F_{17}\! \left(x \right)+F_{37}\! \left(x \right)+F_{39}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{38}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{30}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{4}\! \left(x \right) F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{43}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{37}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{36}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{4}\! \left(x \right) F_{45}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{56}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{47}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{31}\! \left(x \right)+F_{48}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{4}\! \left(x \right) F_{49}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)+F_{53}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{51}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{17}\! \left(x \right)+F_{52}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{4}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{54}\! \left(x \right)\\ F_{54}\! \left(x \right) &= 2 F_{17}\! \left(x \right)+F_{37}\! \left(x \right)+F_{55}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{16}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{57}\! \left(x \right)\\ F_{57}\! \left(x \right) &= 2 F_{17}\! \left(x \right)+F_{37}\! \left(x \right)+F_{58}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{4}\! \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_{12}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{33}\! \left(x \right)\\ \end{align*}\)