Av(12345, 12354, 12435, 12453, 12534, 12543, 13452, 13542, 21345, 21354, 21435, 21453, 21534, 21543, 23451, 23541, 31452, 31542, 32451, 32541)
Generating Function
\(\displaystyle \frac{4 x^{4}-6 x^{3}+14 x^{2}-7 x +1}{\left(2 x^{2}-4 x +1\right)^{2}}\)
Counting Sequence
1, 1, 2, 6, 24, 100, 408, 1624, 6336, 24336, 92320, 346720, 1291392, 4776512, 17562496, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(2 x^{2}-4 x +1\right)^{2} F \! \left(x \right)-4 x^{4}+6 x^{3}-14 x^{2}+7 x -1 = 0\)
Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a{\left(n + 4 \right)} = - 4 a{\left(n \right)} + 16 a{\left(n + 1 \right)} - 20 a{\left(n + 2 \right)} + 8 a{\left(n + 3 \right)}, \quad n \geq 5\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a{\left(n + 4 \right)} = - 4 a{\left(n \right)} + 16 a{\left(n + 1 \right)} - 20 a{\left(n + 2 \right)} + 8 a{\left(n + 3 \right)}, \quad n \geq 5\)
Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0 \\ -\frac{n \left(2 \left(1-\frac{\sqrt{2}}{2}\right)^{-n} \sqrt{2}-2 \left(1+\frac{\sqrt{2}}{2}\right)^{-n} \sqrt{2}-3 \left(1-\frac{\sqrt{2}}{2}\right)^{-n}-3 \left(1+\frac{\sqrt{2}}{2}\right)^{-n}\right)}{4} & \text{otherwise} \end{array}\right.\)
This specification was found using the strategy pack "Regular Insertion Encoding Bottom" and has 68 rules.
Finding the specification took 89 seconds.
Copy 68 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_{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_{38}\! \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_{22}\! \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_{19}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{18}\! \left(x \right) &= 0\\
F_{19}\! \left(x \right) &= F_{11}\! \left(x \right) F_{16}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{16}\! \left(x \right) F_{21}\! \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_{29}\! \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_{22}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{33}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{28}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{16}\! \left(x \right) F_{6}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{16}\! \left(x \right) F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{34}\! \left(x \right)+F_{35}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{16}\! \left(x \right) F_{22}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{16}\! \left(x \right) F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{26}\! \left(x \right)\\
F_{37}\! \left(x \right) &= 0\\
F_{38}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{39}\! \left(x \right)+F_{64}\! \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_{2}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{61}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{44}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{16}\! \left(x \right) F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{16}\! \left(x \right) F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{46}\! \left(x \right)+F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{16}\! \left(x \right) F_{2}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{53}\! \left(x \right)+F_{54}\! \left(x \right)+F_{57}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{16}\! \left(x \right) F_{38}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{16}\! \left(x \right) F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{49}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{16}\! \left(x \right) F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{59}\! \left(x \right)+F_{60}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{17}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{33}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{54}\! \left(x \right)+F_{57}\! \left(x \right)+F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{16}\! \left(x \right) F_{63}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{38}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{16}\! \left(x \right) F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{66}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{61}\! \left(x \right)\\
\end{align*}\)