Av(1342, 1432, 2413, 2431, 3412)
View Raw Data
Generating Function
\(\displaystyle \frac{x^{5}-2 x^{4}+2 x^{3}-7 x^{2}+5 x -1}{\left(x^{2}-3 x +1\right) \left(x^{3}-x^{2}+3 x -1\right)}\)
Copy to clipboard: Search on PermPAL
Counting Sequence
1, 1, 2, 6, 19, 58, 174, 517, 1524, 4462, 12989, 37626, 108532, 311905, 893466, ...
Search on OEIS Search on PermPAL
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{2}-3 x +1\right) \left(x^{3}-x^{2}+3 x -1\right) F \! \left(x \right)-x^{5}+2 x^{4}-2 x^{3}+7 x^{2}-5 x +1 = 0\)
Copy to clipboard: Search on PermPAL
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 58\)
\(\displaystyle a \! \left(n +5\right) = a \! \left(n \right)-4 a \! \left(n +1\right)+7 a \! \left(n +2\right)-11 a \! \left(n +3\right)+6 a \! \left(n +4\right), \quad n \geq 6\)
Copy to clipboard:
Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(\left(\left(-1520 \,\mathrm{I}-240 \sqrt{19}\right) \sqrt{3}-720 \,\mathrm{I} \sqrt{19}-1520\right) \left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{1}{3}}+12160+\left(\left(665 \,\mathrm{I}+75 \sqrt{19}\right) \sqrt{3}-225 \,\mathrm{I} \sqrt{19}-665\right) \left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{\left(\left(\mathrm{I}+3 \sqrt{19}\right) \sqrt{3}-9 \,\mathrm{I} \sqrt{19}-1\right) \left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{2}{3}}}{384}-\frac{\mathrm{I} \sqrt{3}\, \left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{\left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}+\frac{1}{3}\right)^{-n}}{36480}\\+\\\frac{\left(\left(\left(-665 \,\mathrm{I}+75 \sqrt{19}\right) \sqrt{3}+225 \,\mathrm{I} \sqrt{19}-665\right) \left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{2}{3}}+12160+\left(\left(1520 \,\mathrm{I}-240 \sqrt{19}\right) \sqrt{3}+720 \,\mathrm{I} \sqrt{19}-1520\right) \left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{1}{3}}\right) \left(\frac{\left(\left(-\mathrm{I}+3 \sqrt{19}\right) \sqrt{3}+9 \,\mathrm{I} \sqrt{19}-1\right) \left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{2}{3}}}{384}+\frac{\mathrm{I} \sqrt{3}\, \left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{\left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}+\frac{1}{3}\right)^{-n}}{36480}\\+\\\frac{\left(\left(-150 \sqrt{19}\, \sqrt{3}+1330\right) \left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{2}{3}}+480 \left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{1}{3}} \sqrt{19}\, \sqrt{3}+3040 \left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{1}{3}}+12160\right) \left(\frac{\left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{1}{3}}}{3}+\frac{1}{3}+\frac{\left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{2}{3}}}{192}-\frac{\left(1+3 \sqrt{19}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{19}\, \sqrt{3}}{64}\right)^{-n}}{36480}\\+\frac{\left(-3648 \sqrt{5}-18240\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{36480}+\frac{\left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n} \left(\sqrt{5}-5\right)}{10} & \text{otherwise} \end{array}\right.\)
Copy to clipboard:

This specification was found using the strategy pack "Point Placements" and has 27 rules.

Found on July 23, 2021.

Finding the specification took 3 seconds.

Copy to clipboard:

View tree on standalone page.

+ Expand

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