Av(12345, 12354, 12435, 12453, 12534, 12543, 13245, 13254, 13425, 13524, 14235, 14325, 21345, 21354, 21435, 21453, 21534, 21543, 23415, 23514, 24315, 31425, 31524, 32415, 32514)
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Generating Function
\(\displaystyle \frac{x^{6}-2 x^{4}-x^{3}-x^{2}-3 x +1}{x^{6}-x^{5}-4 x^{4}+x^{2}-4 x +1}\)
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Counting Sequence
1, 1, 2, 6, 24, 95, 365, 1390, 5295, 20188, 76988, 293594, 1119593, 4269435, 16280992, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(x^{6}-x^{5}-4 x^{4}+x^{2}-4 x +1\right) F \! \left(x \right)-x^{6}+2 x^{4}+x^{3}+x^{2}+3 x -1 = 0\)
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Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 95\)
\(\displaystyle a(6) = 365\)
\(\displaystyle a{\left(n + 2 \right)} = \frac{a{\left(n \right)}}{4} - \frac{a{\left(n + 1 \right)}}{4} + \frac{a{\left(n + 4 \right)}}{4} - a{\left(n + 5 \right)} + \frac{a{\left(n + 6 \right)}}{4}, \quad n \geq 7\)
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Explicit Closed Form
\(\displaystyle -\frac{815976 \mathit{RootOf} \left(Z^{6}-Z^{5}-4 Z^{4}+Z^{2}-4 Z +1, \mathit{index} =1\right)^{-n +4}}{40991653}-\frac{815976 \mathit{RootOf} \left(Z^{6}-Z^{5}-4 Z^{4}+Z^{2}-4 Z +1, \mathit{index} =2\right)^{-n +4}}{40991653}-\frac{815976 \mathit{RootOf} \left(Z^{6}-Z^{5}-4 Z^{4}+Z^{2}-4 Z +1, \mathit{index} =3\right)^{-n +4}}{40991653}-\frac{815976 \mathit{RootOf} \left(Z^{6}-Z^{5}-4 Z^{4}+Z^{2}-4 Z +1, \mathit{index} =4\right)^{-n +4}}{40991653}-\frac{815976 \mathit{RootOf} \left(Z^{6}-Z^{5}-4 Z^{4}+Z^{2}-4 Z +1, \mathit{index} =5\right)^{-n +4}}{40991653}-\frac{815976 \mathit{RootOf} \left(Z^{6}-Z^{5}-4 Z^{4}+Z^{2}-4 Z +1, \mathit{index} =6\right)^{-n +4}}{40991653}-\frac{1791918 \mathit{RootOf} \left(Z^{6}-Z^{5}-4 Z^{4}+Z^{2}-4 Z +1, \mathit{index} =1\right)^{-n +3}}{40991653}-\frac{1791918 \mathit{RootOf} \left(Z^{6}-Z^{5}-4 Z^{4}+Z^{2}-4 Z +1, \mathit{index} =2\right)^{-n +3}}{40991653}-\frac{1791918 \mathit{RootOf} \left(Z^{6}-Z^{5}-4 Z^{4}+Z^{2}-4 Z +1, \mathit{index} =3\right)^{-n +3}}{40991653}-\frac{1791918 \mathit{RootOf} \left(Z^{6}-Z^{5}-4 Z^{4}+Z^{2}-4 Z +1, \mathit{index} =4\right)^{-n +3}}{40991653}-\frac{1791918 \mathit{RootOf} \left(Z^{6}-Z^{5}-4 Z^{4}+Z^{2}-4 Z +1, \mathit{index} =5\right)^{-n +3}}{40991653}-\frac{1791918 \mathit{RootOf} \left(Z^{6}-Z^{5}-4 Z^{4}+Z^{2}-4 Z +1, \mathit{index} =6\right)^{-n +3}}{40991653}+\frac{3864829 \mathit{RootOf} \left(Z^{6}-Z^{5}-4 Z^{4}+Z^{2}-4 Z +1, \mathit{index} =1\right)^{-n +2}}{40991653}+\frac{3864829 \mathit{RootOf} \left(Z^{6}-Z^{5}-4 Z^{4}+Z^{2}-4 Z +1, \mathit{index} =2\right)^{-n +2}}{40991653}+\frac{3864829 \mathit{RootOf} \left(Z^{6}-Z^{5}-4 Z^{4}+Z^{2}-4 Z +1, \mathit{index} =3\right)^{-n +2}}{40991653}+\frac{3864829 \mathit{RootOf} \left(Z^{6}-Z^{5}-4 Z^{4}+Z^{2}-4 Z +1, \mathit{index} =4\right)^{-n +2}}{40991653}+\frac{3864829 \mathit{RootOf} \left(Z^{6}-Z^{5}-4 Z^{4}+Z^{2}-4 Z +1, \mathit{index} =5\right)^{-n +2}}{40991653}+\frac{3864829 \mathit{RootOf} \left(Z^{6}-Z^{5}-4 Z^{4}+Z^{2}-4 Z +1, \mathit{index} =6\right)^{-n +2}}{40991653}+\frac{9126683 \mathit{RootOf} \left(Z^{6}-Z^{5}-4 Z^{4}+Z^{2}-4 Z +1, \mathit{index} =1\right)^{-n +1}}{40991653}+\frac{9126683 \mathit{RootOf} \left(Z^{6}-Z^{5}-4 Z^{4}+Z^{2}-4 Z +1, \mathit{index} =2\right)^{-n +1}}{40991653}+\frac{9126683 \mathit{RootOf} \left(Z^{6}-Z^{5}-4 Z^{4}+Z^{2}-4 Z +1, \mathit{index} =3\right)^{-n +1}}{40991653}+\frac{9126683 \mathit{RootOf} \left(Z^{6}-Z^{5}-4 Z^{4}+Z^{2}-4 Z +1, \mathit{index} =4\right)^{-n +1}}{40991653}+\frac{9126683 \mathit{RootOf} \left(Z^{6}-Z^{5}-4 Z^{4}+Z^{2}-4 Z +1, \mathit{index} =5\right)^{-n +1}}{40991653}+\frac{9126683 \mathit{RootOf} \left(Z^{6}-Z^{5}-4 Z^{4}+Z^{2}-4 Z +1, \mathit{index} =6\right)^{-n +1}}{40991653}-\frac{144188 \mathit{RootOf} \left(Z^{6}-Z^{5}-4 Z^{4}+Z^{2}-4 Z +1, \mathit{index} =1\right)^{-n -1}}{40991653}-\frac{144188 \mathit{RootOf} \left(Z^{6}-Z^{5}-4 Z^{4}+Z^{2}-4 Z +1, \mathit{index} =2\right)^{-n -1}}{40991653}-\frac{144188 \mathit{RootOf} \left(Z^{6}-Z^{5}-4 Z^{4}+Z^{2}-4 Z +1, \mathit{index} =3\right)^{-n -1}}{40991653}-\frac{144188 \mathit{RootOf} \left(Z^{6}-Z^{5}-4 Z^{4}+Z^{2}-4 Z +1, \mathit{index} =4\right)^{-n -1}}{40991653}-\frac{144188 \mathit{RootOf} \left(Z^{6}-Z^{5}-4 Z^{4}+Z^{2}-4 Z +1, \mathit{index} =5\right)^{-n -1}}{40991653}-\frac{144188 \mathit{RootOf} \left(Z^{6}-Z^{5}-4 Z^{4}+Z^{2}-4 Z +1, \mathit{index} =6\right)^{-n -1}}{40991653}+\frac{2780077 \mathit{RootOf} \left(Z^{6}-Z^{5}-4 Z^{4}+Z^{2}-4 Z +1, \mathit{index} =1\right)^{-n}}{40991653}+\frac{2780077 \mathit{RootOf} \left(Z^{6}-Z^{5}-4 Z^{4}+Z^{2}-4 Z +1, \mathit{index} =2\right)^{-n}}{40991653}+\frac{2780077 \mathit{RootOf} \left(Z^{6}-Z^{5}-4 Z^{4}+Z^{2}-4 Z +1, \mathit{index} =3\right)^{-n}}{40991653}+\frac{2780077 \mathit{RootOf} \left(Z^{6}-Z^{5}-4 Z^{4}+Z^{2}-4 Z +1, \mathit{index} =4\right)^{-n}}{40991653}+\frac{2780077 \mathit{RootOf} \left(Z^{6}-Z^{5}-4 Z^{4}+Z^{2}-4 Z +1, \mathit{index} =5\right)^{-n}}{40991653}+\frac{2780077 \mathit{RootOf} \left(Z^{6}-Z^{5}-4 Z^{4}+Z^{2}-4 Z +1, \mathit{index} =6\right)^{-n}}{40991653}+\left(\left\{\begin{array}{cc}1 & n =0 \\ 0 & \text{otherwise} \end{array}\right.\right)\)
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This specification was found using the strategy pack "Regular Insertion Encoding Bottom" and has 91 rules.

Finding the specification took 109 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_{16}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{16}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{19}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{15}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{17}\! \left(x \right)\\ F_{16}\! \left(x \right) &= x\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{11}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{21}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{20}\! \left(x \right) &= 0\\ F_{21}\! \left(x \right) &= F_{16}\! \left(x \right) F_{22}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{6}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{16}\! \left(x \right) F_{24}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{26}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{16}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{17}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{28}\! \left(x \right)+F_{85}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{16}\! \left(x \right) F_{29}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{31}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{82}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{33}\! \left(x \right)+F_{39}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{16}\! \left(x \right) F_{34}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{36}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{63}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{38}\! \left(x \right)+F_{39}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{16}\! \left(x \right) F_{2}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{16}\! \left(x \right) F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{42}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{37}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{53}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{44}\! \left(x \right)+F_{45}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{16}\! \left(x \right) F_{6}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{16}\! \left(x \right) F_{46}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{49}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{48}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{23}\! \left(x \right)+F_{44}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{20}\! \left(x \right)+F_{51}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{16}\! \left(x \right) F_{52}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{54}\! \left(x \right)+F_{55}\! \left(x \right)+F_{79}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{16}\! \left(x \right) F_{27}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{16}\! \left(x \right) F_{56}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)+F_{75}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{58}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{54}\! \left(x \right)+F_{59}\! \left(x \right)+F_{65}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{16}\! \left(x \right) F_{60}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{61}\! \left(x \right)+F_{62}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{37}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{63}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{64}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{16}\! \left(x \right) F_{32}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{16}\! \left(x \right) F_{66}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{67}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{68}\! \left(x \right)+F_{71}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{20}\! \left(x \right)+F_{69}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{16}\! \left(x \right) F_{70}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{16}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{64}\! \left(x \right)+F_{72}\! \left(x \right)+F_{74}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{16}\! \left(x \right) F_{73}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{37}\! \left(x \right)\\ F_{74}\! \left(x \right) &= 0\\ F_{75}\! \left(x \right) &= F_{76}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{64}\! \left(x \right)+F_{74}\! \left(x \right)+F_{77}\! \left(x \right)\\ F_{77}\! \left(x \right) &= F_{16}\! \left(x \right) F_{78}\! \left(x \right)\\ F_{78}\! \left(x \right) &= F_{36}\! \left(x \right)\\ F_{79}\! \left(x \right) &= F_{16}\! \left(x \right) F_{80}\! \left(x \right)\\ F_{80}\! \left(x \right) &= F_{81}\! \left(x \right)\\ F_{81}\! \left(x \right) &= F_{50}\! \left(x \right)+F_{76}\! \left(x \right)\\ F_{82}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{59}\! \left(x \right)+F_{65}\! \left(x \right)+F_{83}\! \left(x \right)\\ F_{83}\! \left(x \right) &= F_{16}\! \left(x \right) F_{84}\! \left(x \right)\\ F_{84}\! \left(x \right) &= F_{27}\! \left(x \right)\\ F_{85}\! \left(x \right) &= F_{16}\! \left(x \right) F_{86}\! \left(x \right)\\ F_{86}\! \left(x \right) &= F_{87}\! \left(x \right)+F_{88}\! \left(x \right)\\ F_{87}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{88}\! \left(x \right) &= F_{89}\! \left(x \right)+F_{90}\! \left(x \right)\\ F_{89}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{21}\! \left(x \right)+F_{45}\! \left(x \right)\\ F_{90}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{55}\! \left(x \right)+F_{79}\! \left(x \right)+F_{83}\! \left(x \right)\\ \end{align*}\)