Av(132, 1234, 2134, 3241)
View Raw Data
Generating Function
\(\displaystyle \frac{x^{5}+2 x^{4}+x^{3}-x^{2}-x +1}{\left(x^{2}+x -1\right) \left(x^{3}+x^{2}+x -1\right)}\)
Copy to clipboard: Search on PermPAL
Counting Sequence
1, 1, 2, 5, 11, 23, 47, 94, 185, 360, 694, 1328, 2526, 4781, 9012, ...
Search on OEIS Search on PermPAL
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{2}+x -1\right) \left(x^{3}+x^{2}+x -1\right) F \! \left(x \right)-x^{5}-2 x^{4}-x^{3}+x^{2}+x -1 = 0\)
Copy to clipboard: Search on PermPAL
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 5\)
\(\displaystyle a \! \left(4\right) = 11\)
\(\displaystyle a \! \left(5\right) = 23\)
\(\displaystyle a \! \left(n +5\right) = -a \! \left(n \right)-2 a \! \left(n +1\right)-a \! \left(n +2\right)+a \! \left(n +3\right)+2 a \! \left(n +4\right), \quad n \geq 6\)
Copy to clipboard:
Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(\left(\left(550 i-150 \sqrt{11}\right) \sqrt{3}-450 i \sqrt{11}+550\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}+1760+\left(\left(1375 i+225 \sqrt{11}\right) \sqrt{3}-675 i \sqrt{11}-1375\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{\left(\left(17 i+3 \sqrt{11}\right) \sqrt{3}-9 i \sqrt{11}-17\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{24}-\frac{i \sqrt{3}\, \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{1}{3}\right)^{-n}}{2640}\\+\\\frac{\left(\left(\left(-1375 i+225 \sqrt{11}\right) \sqrt{3}+675 i \sqrt{11}-1375\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}+1760+\left(\left(-550 i-150 \sqrt{11}\right) \sqrt{3}+450 i \sqrt{11}+550\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}\right) \left(\frac{\left(\left(-17 i+3 \sqrt{11}\right) \sqrt{3}+9 i \sqrt{11}-17\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{24}+\frac{i \sqrt{3}\, \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{1}{3}\right)^{-n}}{2640}\\+\\\frac{\left(\left(-450 \sqrt{11}\, \sqrt{3}+2750\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}+300 \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}} \sqrt{11}\, \sqrt{3}-1100 \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}+1760\right) \left(\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{3}-\frac{1}{3}+\frac{17 \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{12}-\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{11}\, \sqrt{3}}{4}\right)^{-n}}{2640}\\+\frac{\left(1056 \sqrt{5}-2640\right) \left(-\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{2640}+\frac{\left(-1056 \sqrt{5}-2640\right) \left(\frac{\sqrt{5}}{2}-\frac{1}{2}\right)^{-n}}{2640} & \text{otherwise} \end{array}\right.\)
Copy to clipboard:

This specification was found using the strategy pack "Point Placements" and has 28 rules.

Found on January 18, 2022.

Finding the specification took 1 seconds.

Copy to clipboard:

View tree on standalone page.

+ Expand

Copy 28 equations to clipboard:
\(\begin{align*} F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{1}\! \left(x \right) &= 1\\ F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{4}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{4}\! \left(x \right) &= x\\ F_{5}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{4}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{13}\! \left(x \right)+F_{24}\! \left(x \right)\\ F_{12}\! \left(x \right) &= 0\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{19}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{4}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{21}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{13}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{20}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{13}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{2}\! \left(x \right) F_{4}\! \left(x \right)\\ \end{align*}\)