Av(1342, 1423, 2143, 2314)
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Generating Function
\(\displaystyle -\frac{\left(x -1\right) \left(2 x -1\right)^{2}}{2 x^{5}+2 x^{4}-10 x^{3}+12 x^{2}-6 x +1}\)
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Counting Sequence
1, 1, 2, 6, 20, 64, 198, 604, 1836, 5580, 16964, 51580, 156832, 476840, 1449768, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(2 x^{5}+2 x^{4}-10 x^{3}+12 x^{2}-6 x +1\right) F \! \left(x \right)+\left(x -1\right) \left(2 x -1\right)^{2} = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(n +5\right) = -2 a \! \left(n \right)-2 a \! \left(n +1\right)+10 a \! \left(n +2\right)-12 a \! \left(n +3\right)+6 a \! \left(n +4\right), \quad n \geq 5\)
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Explicit Closed Form
\(\displaystyle -\frac{974 \mathit{RootOf} \left(2 Z^{5}+2 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =1\right)^{-n +3}}{2059}-\frac{974 \mathit{RootOf} \left(2 Z^{5}+2 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =2\right)^{-n +3}}{2059}-\frac{974 \mathit{RootOf} \left(2 Z^{5}+2 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)^{-n +3}}{2059}-\frac{974 \mathit{RootOf} \left(2 Z^{5}+2 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =4\right)^{-n +3}}{2059}-\frac{974 \mathit{RootOf} \left(2 Z^{5}+2 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =5\right)^{-n +3}}{2059}-\frac{60 \mathit{RootOf} \left(2 Z^{5}+2 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =1\right)^{-n +2}}{71}-\frac{60 \mathit{RootOf} \left(2 Z^{5}+2 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =2\right)^{-n +2}}{71}-\frac{60 \mathit{RootOf} \left(2 Z^{5}+2 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)^{-n +2}}{71}-\frac{60 \mathit{RootOf} \left(2 Z^{5}+2 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =4\right)^{-n +2}}{71}-\frac{60 \mathit{RootOf} \left(2 Z^{5}+2 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =5\right)^{-n +2}}{71}+\frac{3565 \mathit{RootOf} \left(2 Z^{5}+2 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =1\right)^{-n +1}}{2059}+\frac{3565 \mathit{RootOf} \left(2 Z^{5}+2 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =2\right)^{-n +1}}{2059}+\frac{3565 \mathit{RootOf} \left(2 Z^{5}+2 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)^{-n +1}}{2059}+\frac{3565 \mathit{RootOf} \left(2 Z^{5}+2 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =4\right)^{-n +1}}{2059}+\frac{3565 \mathit{RootOf} \left(2 Z^{5}+2 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =5\right)^{-n +1}}{2059}+\frac{673 \mathit{RootOf} \left(2 Z^{5}+2 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =1\right)^{-n -1}}{2059}+\frac{673 \mathit{RootOf} \left(2 Z^{5}+2 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =2\right)^{-n -1}}{2059}+\frac{673 \mathit{RootOf} \left(2 Z^{5}+2 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)^{-n -1}}{2059}+\frac{673 \mathit{RootOf} \left(2 Z^{5}+2 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =4\right)^{-n -1}}{2059}+\frac{673 \mathit{RootOf} \left(2 Z^{5}+2 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =5\right)^{-n -1}}{2059}-\frac{2478 \mathit{RootOf} \left(2 Z^{5}+2 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =1\right)^{-n}}{2059}-\frac{2478 \mathit{RootOf} \left(2 Z^{5}+2 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =2\right)^{-n}}{2059}-\frac{2478 \mathit{RootOf} \left(2 Z^{5}+2 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =3\right)^{-n}}{2059}-\frac{2478 \mathit{RootOf} \left(2 Z^{5}+2 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =4\right)^{-n}}{2059}-\frac{2478 \mathit{RootOf} \left(2 Z^{5}+2 Z^{4}-10 Z^{3}+12 Z^{2}-6 Z +1, \mathit{index} =5\right)^{-n}}{2059}\)
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This specification was found using the strategy pack "Point Placements" and has 57 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_{13}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{21}\! \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_{13}\! \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_{14}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right) F_{13}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x \right)\\ F_{13}\! \left(x \right) &= x\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{13}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{20}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{13}\! \left(x \right) F_{19}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{17}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{22}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{24}\! \left(x \right)+F_{50}\! \left(x \right)\\ F_{23}\! \left(x \right) &= 0\\ F_{24}\! \left(x \right) &= F_{13}\! \left(x \right) F_{25}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{13}\! \left(x \right) F_{29}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{10}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{34}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{24}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{13}\! \left(x \right) F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{31}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{35}\! \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_{37}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{38}\! \left(x \right)+F_{48}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{13}\! \left(x \right) F_{39}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{43}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{10}\! \left(x \right) F_{13}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{46}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{38}\! \left(x \right)+F_{45}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{13}\! \left(x \right) F_{2}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{13}\! \left(x \right) F_{37}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{13}\! \left(x \right) F_{49}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{37}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{13}\! \left(x \right) F_{51}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{52}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{53}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{13}\! \left(x \right) F_{55}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{56}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{54}\! \left(x \right)\\ \end{align*}\)