Av(1342, 1423, 2143, 2314, 3124)
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Generating Function
\(\displaystyle -\frac{\left(2 x -1\right) \left(x -1\right)^{3}}{2 x^{6}-2 x^{5}-7 x^{4}+14 x^{3}-13 x^{2}+6 x -1}\)
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Counting Sequence
1, 1, 2, 6, 19, 55, 153, 425, 1190, 3346, 9413, 26469, 74415, 209223, 588306, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(2 x^{6}-2 x^{5}-7 x^{4}+14 x^{3}-13 x^{2}+6 x -1\right) F \! \left(x \right)+\left(2 x -1\right) \left(x -1\right)^{3} = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 55\)
\(\displaystyle a \! \left(n +6\right) = 2 a \! \left(n \right)-2 a \! \left(n +1\right)-7 a \! \left(n +2\right)+14 a \! \left(n +3\right)-13 a \! \left(n +4\right)+6 a \! \left(n +5\right), \quad n \geq 6\)
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Explicit Closed Form
\(\displaystyle \frac{573 \mathit{RootOf} \left(2 Z^{6}-2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =1\right)^{-n +4}}{1559}+\frac{573 \mathit{RootOf} \left(2 Z^{6}-2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =2\right)^{-n +4}}{1559}+\frac{573 \mathit{RootOf} \left(2 Z^{6}-2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =3\right)^{-n +4}}{1559}+\frac{573 \mathit{RootOf} \left(2 Z^{6}-2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =4\right)^{-n +4}}{1559}+\frac{573 \mathit{RootOf} \left(2 Z^{6}-2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =5\right)^{-n +4}}{1559}+\frac{573 \mathit{RootOf} \left(2 Z^{6}-2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =6\right)^{-n +4}}{1559}-\frac{854 \mathit{RootOf} \left(2 Z^{6}-2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =1\right)^{-n +3}}{4677}-\frac{854 \mathit{RootOf} \left(2 Z^{6}-2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =2\right)^{-n +3}}{4677}-\frac{854 \mathit{RootOf} \left(2 Z^{6}-2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =3\right)^{-n +3}}{4677}-\frac{854 \mathit{RootOf} \left(2 Z^{6}-2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =4\right)^{-n +3}}{4677}-\frac{854 \mathit{RootOf} \left(2 Z^{6}-2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =5\right)^{-n +3}}{4677}-\frac{854 \mathit{RootOf} \left(2 Z^{6}-2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =6\right)^{-n +3}}{4677}-\frac{13277 \mathit{RootOf} \left(2 Z^{6}-2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =1\right)^{-n +2}}{9354}-\frac{13277 \mathit{RootOf} \left(2 Z^{6}-2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =2\right)^{-n +2}}{9354}-\frac{13277 \mathit{RootOf} \left(2 Z^{6}-2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =3\right)^{-n +2}}{9354}-\frac{13277 \mathit{RootOf} \left(2 Z^{6}-2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =4\right)^{-n +2}}{9354}-\frac{13277 \mathit{RootOf} \left(2 Z^{6}-2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =5\right)^{-n +2}}{9354}-\frac{13277 \mathit{RootOf} \left(2 Z^{6}-2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =6\right)^{-n +2}}{9354}+\frac{16937 \mathit{RootOf} \left(2 Z^{6}-2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =1\right)^{-n +1}}{9354}+\frac{16937 \mathit{RootOf} \left(2 Z^{6}-2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =2\right)^{-n +1}}{9354}+\frac{16937 \mathit{RootOf} \left(2 Z^{6}-2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =3\right)^{-n +1}}{9354}+\frac{16937 \mathit{RootOf} \left(2 Z^{6}-2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =4\right)^{-n +1}}{9354}+\frac{16937 \mathit{RootOf} \left(2 Z^{6}-2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =5\right)^{-n +1}}{9354}+\frac{16937 \mathit{RootOf} \left(2 Z^{6}-2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =6\right)^{-n +1}}{9354}+\frac{3463 \mathit{RootOf} \left(2 Z^{6}-2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =1\right)^{-n -1}}{9354}+\frac{3463 \mathit{RootOf} \left(2 Z^{6}-2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =2\right)^{-n -1}}{9354}+\frac{3463 \mathit{RootOf} \left(2 Z^{6}-2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =3\right)^{-n -1}}{9354}+\frac{3463 \mathit{RootOf} \left(2 Z^{6}-2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =4\right)^{-n -1}}{9354}+\frac{3463 \mathit{RootOf} \left(2 Z^{6}-2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =5\right)^{-n -1}}{9354}+\frac{3463 \mathit{RootOf} \left(2 Z^{6}-2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =6\right)^{-n -1}}{9354}-\frac{5608 \mathit{RootOf} \left(2 Z^{6}-2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =1\right)^{-n}}{4677}-\frac{5608 \mathit{RootOf} \left(2 Z^{6}-2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =2\right)^{-n}}{4677}-\frac{5608 \mathit{RootOf} \left(2 Z^{6}-2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =3\right)^{-n}}{4677}-\frac{5608 \mathit{RootOf} \left(2 Z^{6}-2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =4\right)^{-n}}{4677}-\frac{5608 \mathit{RootOf} \left(2 Z^{6}-2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =5\right)^{-n}}{4677}-\frac{5608 \mathit{RootOf} \left(2 Z^{6}-2 Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =6\right)^{-n}}{4677}\)
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This specification was found using the strategy pack "Point Placements" and has 70 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_{21}\! \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_{14}\! \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_{1}\! \left(x \right)+F_{11}\! \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_{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_{19}\! \left(x \right) F_{4}\! \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_{51}\! \left(x \right)\\ F_{23}\! \left(x \right) &= 0\\ 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_{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_{29}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{11}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{35}\! \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_{33}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{34}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{32}\! \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_{38}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{39}\! \left(x \right)+F_{49}\! \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_{44}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{11}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{47}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{39}\! \left(x \right)+F_{46}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{2}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{38}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{4}\! \left(x \right) F_{50}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{38}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{4}\! \left(x \right) F_{52}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)+F_{65}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{54}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{4}\! \left(x \right) F_{56}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{53}\! \left(x \right)+F_{57}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)+F_{61}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{59}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{4}\! \left(x \right) F_{60}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{58}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{62}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{4}\! \left(x \right) F_{63}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{64}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{62}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{66}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{67}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{4}\! \left(x \right) F_{68}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{69}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{67}\! \left(x \right)\\ \end{align*}\)