Av(132, 1234, 2314, 3241)
View Raw Data
Generating Function
\(\displaystyle \frac{x^{5}-x^{4}-x^{3}-x^{2}+2 x -1}{\left(x^{3}+x^{2}+x -1\right) \left(x -1\right)^{2}}\)
Copy to clipboard: Search on PermPAL
Counting Sequence
1, 1, 2, 5, 11, 22, 43, 82, 154, 287, 532, 983, 1813, 3340, 6149, ...
Search on OEIS Search on PermPAL
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{3}+x^{2}+x -1\right) \left(x -1\right)^{2} F \! \left(x \right)-x^{5}+x^{4}+x^{3}+x^{2}-2 x +1 = 0\)
Copy to clipboard: Search on PermPAL
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 5\)
\(\displaystyle a \! \left(4\right) = 11\)
\(\displaystyle a \! \left(5\right) = 22\)
\(\displaystyle a \! \left(n +3\right) = a \! \left(n \right)+a \! \left(n +1\right)+a \! \left(n +2\right)+n +2, \quad n \geq 6\)
Copy to clipboard:
Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(\left(\left(55 \,\mathrm{I}-17 \sqrt{11}\right) \sqrt{3}-51 \,\mathrm{I} \sqrt{11}+55\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}+176+\left(\left(187 \,\mathrm{I}+31 \sqrt{11}\right) \sqrt{3}-93 \,\mathrm{I} \sqrt{11}-187\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{\left(\left(17 \,\mathrm{I}+3 \sqrt{11}\right) \sqrt{3}-9 \,\mathrm{I} \sqrt{11}-17\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{24}-\frac{\mathrm{I} \sqrt{3}\, \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{1}{3}\right)^{-n}}{528}\\+\\\frac{\left(\left(\left(-187 \,\mathrm{I}+31 \sqrt{11}\right) \sqrt{3}+93 \,\mathrm{I} \sqrt{11}-187\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}+176+\left(\left(-55 \,\mathrm{I}-17 \sqrt{11}\right) \sqrt{3}+51 \,\mathrm{I} \sqrt{11}+55\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}\right) \left(\frac{\left(\left(-17 \,\mathrm{I}+3 \sqrt{11}\right) \sqrt{3}+9 \,\mathrm{I} \sqrt{11}-17\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{24}+\frac{\mathrm{I} \sqrt{3}\, \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{1}{3}\right)^{-n}}{528}\\+\\\frac{\left(\left(-62 \sqrt{11}\, \sqrt{3}+374\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}+34 \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}} \sqrt{11}\, \sqrt{3}-110 \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}+176\right) \left(\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{3}-\frac{1}{3}+\frac{17 \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{12}-\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{11}\, \sqrt{3}}{4}\right)^{-n}}{528}\\-\frac{n}{2}-1 & \text{otherwise} \end{array}\right.\)
Copy to clipboard:

This specification was found using the strategy pack "Point Placements" and has 30 rules.

Found on January 18, 2022.

Finding the specification took 1 seconds.

Copy to clipboard:

View tree on standalone page.

+ Expand

Copy 30 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_{4}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{4}\! \left(x \right) &= x\\ F_{5}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{4}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{13}\! \left(x \right)+F_{26}\! \left(x \right)\\ F_{12}\! \left(x \right) &= 0\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{19}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{4}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{21}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{13}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{25}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{16}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{22}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{2}\! \left(x \right) F_{4}\! \left(x \right)\\ \end{align*}\)