Av(1342, 2413, 4123)
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Generating Function
\(\displaystyle -\frac{5 x^{4}-11 x^{3}+12 x^{2}-6 x +1}{2 x^{5}-12 x^{4}+20 x^{3}-17 x^{2}+7 x -1}\)
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Counting Sequence
1, 1, 2, 6, 21, 75, 266, 939, 3311, 11676, 41183, 145273, 512466, 1807791, 6377231, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(2 x^{5}-12 x^{4}+20 x^{3}-17 x^{2}+7 x -1\right) F \! \left(x \right)+5 x^{4}-11 x^{3}+12 x^{2}-6 x +1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(n +5\right) = 2 a \! \left(n \right)-12 a \! \left(n +1\right)+20 a \! \left(n +2\right)-17 a \! \left(n +3\right)+7 a \! \left(n +4\right), \quad n \geq 5\)
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Explicit Closed Form
\(\displaystyle \frac{1654 \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+20 Z^{3}-17 Z^{2}+7 Z -1, \mathit{index} =1\right)^{-n +3}}{9383}+\frac{1654 \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+20 Z^{3}-17 Z^{2}+7 Z -1, \mathit{index} =2\right)^{-n +3}}{9383}+\frac{1654 \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+20 Z^{3}-17 Z^{2}+7 Z -1, \mathit{index} =3\right)^{-n +3}}{9383}+\frac{1654 \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+20 Z^{3}-17 Z^{2}+7 Z -1, \mathit{index} =4\right)^{-n +3}}{9383}+\frac{1654 \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+20 Z^{3}-17 Z^{2}+7 Z -1, \mathit{index} =5\right)^{-n +3}}{9383}-\frac{8995 \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+20 Z^{3}-17 Z^{2}+7 Z -1, \mathit{index} =1\right)^{-n +2}}{9383}-\frac{8995 \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+20 Z^{3}-17 Z^{2}+7 Z -1, \mathit{index} =2\right)^{-n +2}}{9383}-\frac{8995 \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+20 Z^{3}-17 Z^{2}+7 Z -1, \mathit{index} =3\right)^{-n +2}}{9383}-\frac{8995 \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+20 Z^{3}-17 Z^{2}+7 Z -1, \mathit{index} =4\right)^{-n +2}}{9383}-\frac{8995 \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+20 Z^{3}-17 Z^{2}+7 Z -1, \mathit{index} =5\right)^{-n +2}}{9383}+\frac{12889 \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+20 Z^{3}-17 Z^{2}+7 Z -1, \mathit{index} =1\right)^{-n +1}}{9383}+\frac{12889 \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+20 Z^{3}-17 Z^{2}+7 Z -1, \mathit{index} =2\right)^{-n +1}}{9383}+\frac{12889 \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+20 Z^{3}-17 Z^{2}+7 Z -1, \mathit{index} =3\right)^{-n +1}}{9383}+\frac{12889 \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+20 Z^{3}-17 Z^{2}+7 Z -1, \mathit{index} =4\right)^{-n +1}}{9383}+\frac{12889 \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+20 Z^{3}-17 Z^{2}+7 Z -1, \mathit{index} =5\right)^{-n +1}}{9383}+\frac{1634 \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+20 Z^{3}-17 Z^{2}+7 Z -1, \mathit{index} =1\right)^{-n -1}}{9383}+\frac{1634 \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+20 Z^{3}-17 Z^{2}+7 Z -1, \mathit{index} =2\right)^{-n -1}}{9383}+\frac{1634 \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+20 Z^{3}-17 Z^{2}+7 Z -1, \mathit{index} =3\right)^{-n -1}}{9383}+\frac{1634 \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+20 Z^{3}-17 Z^{2}+7 Z -1, \mathit{index} =4\right)^{-n -1}}{9383}+\frac{1634 \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+20 Z^{3}-17 Z^{2}+7 Z -1, \mathit{index} =5\right)^{-n -1}}{9383}-\frac{7438 \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+20 Z^{3}-17 Z^{2}+7 Z -1, \mathit{index} =1\right)^{-n}}{9383}-\frac{7438 \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+20 Z^{3}-17 Z^{2}+7 Z -1, \mathit{index} =2\right)^{-n}}{9383}-\frac{7438 \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+20 Z^{3}-17 Z^{2}+7 Z -1, \mathit{index} =3\right)^{-n}}{9383}-\frac{7438 \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+20 Z^{3}-17 Z^{2}+7 Z -1, \mathit{index} =4\right)^{-n}}{9383}-\frac{7438 \mathit{RootOf} \left(2 Z^{5}-12 Z^{4}+20 Z^{3}-17 Z^{2}+7 Z -1, \mathit{index} =5\right)^{-n}}{9383}\)
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This specification was found using the strategy pack "Point Placements" and has 48 rules.

Found on July 23, 2021.

Finding the specification took 7 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_{35}\! \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_{6}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{17}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{14}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{18}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{22}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{18}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{26}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{34}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{27}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{30}\! \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_{38}\! \left(x \right)+F_{39}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{5}\! \left(x \right) F_{6}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{14}\! \left(x \right) F_{4}\! \left(x \right) F_{41}\! \left(x \right) F_{5}\! \left(x \right) F_{6}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{43}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{45}\! \left(x \right)+F_{47}\! \left(x \right)\\ F_{44}\! \left(x \right) &= 0\\ F_{45}\! \left(x \right) &= F_{4}\! \left(x \right) F_{46}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{43}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{4}\! \left(x \right) F_{42}\! \left(x \right)\\ \end{align*}\)