Av(12354, 12453, 12543, 13254, 13452, 13542, 14253, 14352, 14532, 15243, 15342, 15432, 21354, 21453, 21543, 23451, 23541, 24351, 24531, 25341, 25431, 31452, 31542, 32451, 32541)
Generating Function
\(\displaystyle \frac{x^{4}+2 x^{3}+5 x^{2}-5 x +1}{\left(x -1\right) \left(x^{2}-4 x +1\right) \left(x^{2}+x -1\right)}\)
Counting Sequence
1, 1, 2, 6, 24, 95, 367, 1393, 5240, 19626, 73362, 273983, 1022833, 3817777, 14248970, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x -1\right) \left(x^{2}-4 x +1\right) \left(x^{2}+x -1\right) F \! \left(x \right)-x^{4}-2 x^{3}-5 x^{2}+5 x -1 = 0\)
Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a{\left(n + 4 \right)} = a{\left(n \right)} - 3 a{\left(n + 1 \right)} - 4 a{\left(n + 2 \right)} + 5 a{\left(n + 3 \right)} + 4, \quad n \geq 5\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a{\left(n + 4 \right)} = a{\left(n \right)} - 3 a{\left(n + 1 \right)} - 4 a{\left(n + 2 \right)} + 5 a{\left(n + 3 \right)} + 4, \quad n \geq 5\)
Explicit Closed Form
\(\displaystyle \frac{\left(72 \sqrt{5}-90\right) \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{330}+\frac{\left(-72 \sqrt{5}-90\right) \left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{330}+\frac{\left(70 \sqrt{3}-75\right) \left(2-\sqrt{3}\right)^{-n}}{330}+2+\frac{\left(-70 \sqrt{3}-75\right) \left(2+\sqrt{3}\right)^{-n}}{330}\)
This specification was found using the strategy pack "Regular Insertion Encoding Bottom" and has 54 rules.
Finding the specification took 43 seconds.
Copy 54 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_{44}\! \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_{22}\! \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_{16}\! \left(x \right)+F_{21}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{15}\! \left(x \right) F_{2}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{25}\! \left(x \right)+F_{26}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{15}\! \left(x \right) F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{6}\! \left(x \right)\\
F_{25}\! \left(x \right) &= 0\\
F_{26}\! \left(x \right) &= F_{15}\! \left(x \right) F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{15}\! \left(x \right) F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{15}\! \left(x \right) F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= 2 F_{7}\! \left(x \right)+F_{26}\! \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_{40}\! \left(x \right)+F_{44}\! \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_{20}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{15}\! \left(x \right) F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{50}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{48}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{15}\! \left(x \right) F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{26}\! \left(x \right)+F_{52}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{15}\! \left(x \right) F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{6}\! \left(x \right)\\
\end{align*}\)