Av(1243, 1324, 2143, 2431, 3412, 4231)
View Raw Data
Generating Function
\(\displaystyle \frac{2 x^{6}-7 x^{5}+x^{4}-2 x^{3}+6 x^{2}-4 x +1}{\left(2 x -1\right) \left(x -1\right)^{3}}\)
Copy to clipboard: Search on PermPAL
Counting Sequence
1, 1, 2, 6, 18, 41, 83, 160, 304, 579, 1113, 2162, 4238, 8365, 16591, ...
Search on OEIS Search on PermPAL
Implicit Equation for the Generating Function
\(\displaystyle -\left(2 x -1\right) \left(x -1\right)^{3} F \! \left(x \right)+2 x^{6}-7 x^{5}+x^{4}-2 x^{3}+6 x^{2}-4 x +1 = 0\)
Copy to clipboard: Search on PermPAL
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 18\)
\(\displaystyle a \! \left(5\right) = 41\)
\(\displaystyle a \! \left(6\right) = 83\)
\(\displaystyle a \! \left(n +1\right) = -\frac{3 n^{2}}{2}+2 a \! \left(n \right)+\frac{19 n}{2}-9, \quad n \geq 7\)
Copy to clipboard:
Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0 \\ 2 & n =2 \\ 4-\frac{13 n}{2}+\frac{3 n^{2}}{2}+2^{n} & \text{otherwise} \end{array}\right.\)
Copy to clipboard:

This specification was found using the strategy pack "Point Placements" and has 50 rules.

Found on July 23, 2021.

Finding the specification took 7 seconds.

Copy to clipboard:

View tree on standalone page.

+ Expand

Copy 50 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_{24}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{15}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{5}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right) F_{15}\! \left(x \right) F_{22}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{19}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{13}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{12}\! \left(x \right) F_{15}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{15}\! \left(x \right) &= x\\ F_{16}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{17}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{15}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{15}\! \left(x \right) F_{21}\! \left(x \right) F_{22}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{19}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{15}\! \left(x \right) F_{22}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)\\ 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_{40}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{38}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{34}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{33}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{1}\! \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_{28}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{12}\! \left(x \right) F_{17}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{15}\! \left(x \right) F_{36}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{15}\! \left(x \right) F_{16}\! \left(x \right) F_{22}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{15}\! \left(x \right) F_{39}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{36}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{15}\! \left(x \right) F_{22}\! \left(x \right) F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{43}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{16}\! \left(x \right) F_{44}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{45}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{48}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{22}\! \left(x \right) F_{47}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{15}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{15}\! \left(x \right) F_{17}\! \left(x \right) F_{21}\! \left(x \right) F_{22}\! \left(x \right)\\ \end{align*}\)