Av(12453, 12534, 12543, 13452, 13524, 13542, 14352, 14523, 14532, 15342, 15423, 15432, 21453, 21534, 21543, 23451, 23541, 24351, 24531, 25341, 25431, 31452, 31542, 32451, 32541)
Generating Function
\(\displaystyle -\frac{\left(x -1\right) \left(2 x^{4}+4 x^{3}-3 x^{2}-3 x +1\right)}{3 x^{6}-x^{5}-10 x^{4}+8 x^{3}+3 x^{2}-5 x +1}\)
Counting Sequence
1, 1, 2, 6, 24, 95, 373, 1447, 5596, 21611, 83444, 322177, 1243953, 4803047, 18545223, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(3 x^{6}-x^{5}-10 x^{4}+8 x^{3}+3 x^{2}-5 x +1\right) F \! \left(x \right)+\left(x -1\right) \left(2 x^{4}+4 x^{3}-3 x^{2}-3 x +1\right) = 0\)
Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 95\)
\(\displaystyle a{\left(n + 6 \right)} = - 3 a{\left(n \right)} + a{\left(n + 1 \right)} + 10 a{\left(n + 2 \right)} - 8 a{\left(n + 3 \right)} - 3 a{\left(n + 4 \right)} + 5 a{\left(n + 5 \right)}, \quad n \geq 6\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 95\)
\(\displaystyle a{\left(n + 6 \right)} = - 3 a{\left(n \right)} + a{\left(n + 1 \right)} + 10 a{\left(n + 2 \right)} - 8 a{\left(n + 3 \right)} - 3 a{\left(n + 4 \right)} + 5 a{\left(n + 5 \right)}, \quad n \geq 6\)
Explicit Closed Form
\(\displaystyle \frac{27546099 \mathit{RootOf} \left(3 Z^{6}-Z^{5}-10 Z^{4}+8 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =1\right)^{-n +4}}{133073635}+\frac{27546099 \mathit{RootOf} \left(3 Z^{6}-Z^{5}-10 Z^{4}+8 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =2\right)^{-n +4}}{133073635}+\frac{27546099 \mathit{RootOf} \left(3 Z^{6}-Z^{5}-10 Z^{4}+8 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =3\right)^{-n +4}}{133073635}+\frac{27546099 \mathit{RootOf} \left(3 Z^{6}-Z^{5}-10 Z^{4}+8 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =4\right)^{-n +4}}{133073635}+\frac{27546099 \mathit{RootOf} \left(3 Z^{6}-Z^{5}-10 Z^{4}+8 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =5\right)^{-n +4}}{133073635}+\frac{27546099 \mathit{RootOf} \left(3 Z^{6}-Z^{5}-10 Z^{4}+8 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =6\right)^{-n +4}}{133073635}-\frac{1136177 \mathit{RootOf} \left(3 Z^{6}-Z^{5}-10 Z^{4}+8 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =1\right)^{-n +3}}{133073635}-\frac{1136177 \mathit{RootOf} \left(3 Z^{6}-Z^{5}-10 Z^{4}+8 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =2\right)^{-n +3}}{133073635}-\frac{1136177 \mathit{RootOf} \left(3 Z^{6}-Z^{5}-10 Z^{4}+8 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =3\right)^{-n +3}}{133073635}-\frac{1136177 \mathit{RootOf} \left(3 Z^{6}-Z^{5}-10 Z^{4}+8 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =4\right)^{-n +3}}{133073635}-\frac{1136177 \mathit{RootOf} \left(3 Z^{6}-Z^{5}-10 Z^{4}+8 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =5\right)^{-n +3}}{133073635}-\frac{1136177 \mathit{RootOf} \left(3 Z^{6}-Z^{5}-10 Z^{4}+8 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =6\right)^{-n +3}}{133073635}-\frac{98390138 \mathit{RootOf} \left(3 Z^{6}-Z^{5}-10 Z^{4}+8 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =1\right)^{-n +2}}{133073635}-\frac{98390138 \mathit{RootOf} \left(3 Z^{6}-Z^{5}-10 Z^{4}+8 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =2\right)^{-n +2}}{133073635}-\frac{98390138 \mathit{RootOf} \left(3 Z^{6}-Z^{5}-10 Z^{4}+8 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =3\right)^{-n +2}}{133073635}-\frac{98390138 \mathit{RootOf} \left(3 Z^{6}-Z^{5}-10 Z^{4}+8 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =4\right)^{-n +2}}{133073635}-\frac{98390138 \mathit{RootOf} \left(3 Z^{6}-Z^{5}-10 Z^{4}+8 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =5\right)^{-n +2}}{133073635}-\frac{98390138 \mathit{RootOf} \left(3 Z^{6}-Z^{5}-10 Z^{4}+8 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =6\right)^{-n +2}}{133073635}+\frac{40108417 \mathit{RootOf} \left(3 Z^{6}-Z^{5}-10 Z^{4}+8 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =1\right)^{-n +1}}{133073635}+\frac{40108417 \mathit{RootOf} \left(3 Z^{6}-Z^{5}-10 Z^{4}+8 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =2\right)^{-n +1}}{133073635}+\frac{40108417 \mathit{RootOf} \left(3 Z^{6}-Z^{5}-10 Z^{4}+8 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =3\right)^{-n +1}}{133073635}+\frac{40108417 \mathit{RootOf} \left(3 Z^{6}-Z^{5}-10 Z^{4}+8 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =4\right)^{-n +1}}{133073635}+\frac{40108417 \mathit{RootOf} \left(3 Z^{6}-Z^{5}-10 Z^{4}+8 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =5\right)^{-n +1}}{133073635}+\frac{40108417 \mathit{RootOf} \left(3 Z^{6}-Z^{5}-10 Z^{4}+8 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =6\right)^{-n +1}}{133073635}-\frac{14809742 \mathit{RootOf} \left(3 Z^{6}-Z^{5}-10 Z^{4}+8 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =1\right)^{-n -1}}{133073635}-\frac{14809742 \mathit{RootOf} \left(3 Z^{6}-Z^{5}-10 Z^{4}+8 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =2\right)^{-n -1}}{133073635}-\frac{14809742 \mathit{RootOf} \left(3 Z^{6}-Z^{5}-10 Z^{4}+8 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =3\right)^{-n -1}}{133073635}-\frac{14809742 \mathit{RootOf} \left(3 Z^{6}-Z^{5}-10 Z^{4}+8 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =4\right)^{-n -1}}{133073635}-\frac{14809742 \mathit{RootOf} \left(3 Z^{6}-Z^{5}-10 Z^{4}+8 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =5\right)^{-n -1}}{133073635}-\frac{14809742 \mathit{RootOf} \left(3 Z^{6}-Z^{5}-10 Z^{4}+8 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =6\right)^{-n -1}}{133073635}+\frac{1120767 \mathit{RootOf} \left(3 Z^{6}-Z^{5}-10 Z^{4}+8 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =1\right)^{-n}}{2181535}+\frac{1120767 \mathit{RootOf} \left(3 Z^{6}-Z^{5}-10 Z^{4}+8 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =2\right)^{-n}}{2181535}+\frac{1120767 \mathit{RootOf} \left(3 Z^{6}-Z^{5}-10 Z^{4}+8 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =3\right)^{-n}}{2181535}+\frac{1120767 \mathit{RootOf} \left(3 Z^{6}-Z^{5}-10 Z^{4}+8 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =4\right)^{-n}}{2181535}+\frac{1120767 \mathit{RootOf} \left(3 Z^{6}-Z^{5}-10 Z^{4}+8 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =5\right)^{-n}}{2181535}+\frac{1120767 \mathit{RootOf} \left(3 Z^{6}-Z^{5}-10 Z^{4}+8 Z^{3}+3 Z^{2}-5 Z +1, \mathit{index} =6\right)^{-n}}{2181535}\)
This specification was found using the strategy pack "Regular Insertion Encoding Bottom" and has 55 rules.
Finding the specification took 74 seconds.
Copy 55 equations to clipboard:
\(\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_{15}\! \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_{2}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{48}\! \left(x \right)+F_{7}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{7}\! \left(x \right) &= 0\\
F_{8}\! \left(x \right) &= F_{15}\! \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_{2}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{21}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{16}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right) F_{15}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{15}\! \left(x \right) &= x\\
F_{16}\! \left(x \right) &= F_{15}\! \left(x \right) F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{15}\! \left(x \right) F_{2}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{24}\! \left(x \right)+F_{25}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{15}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{6}\! \left(x \right)\\
F_{24}\! \left(x \right) &= 0\\
F_{25}\! \left(x \right) &= F_{15}\! \left(x \right) F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{29}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{15}\! \left(x \right) F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{33}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{20}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{35}\! \left(x \right)+F_{36}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{15}\! \left(x \right) F_{6}\! \left(x \right)\\
F_{36}\! \left(x \right) &= 0\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{15}\! \left(x \right) F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{40}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{15}\! \left(x \right) F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{39}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{13}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{15}\! \left(x \right) F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{39}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{15}\! \left(x \right) F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{36}\! \left(x \right)+F_{53}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{15}\! \left(x \right) F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{6}\! \left(x \right)\\
\end{align*}\)