Av(132, 1234, 2134, 3214)
View Raw Data
Generating Function
\(\displaystyle -\frac{1}{\left(x^{2}+1\right) \left(x^{3}+2 x^{2}+x -1\right)}\)
Copy to clipboard: Search on PermPAL
Counting Sequence
1, 1, 2, 5, 11, 23, 49, 106, 228, 489, 1050, 2256, 4846, 10408, 22355, ...
Search on OEIS Search on PermPAL
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{2}+1\right) \left(x^{3}+2 x^{2}+x -1\right) F \! \left(x \right)+1 = 0\)
Copy to clipboard: Search on PermPAL
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 5\)
\(\displaystyle a \! \left(4\right) = 11\)
\(\displaystyle a \! \left(n +5\right) = a \! \left(n \right)+2 a \! \left(n +1\right)+2 a \! \left(n +2\right)+a \! \left(n +3\right)+a \! \left(n +4\right), \quad n \geq 5\)
Copy to clipboard:
Explicit Closed Form
\(\displaystyle -\frac{949 \left(2^{\frac{1}{3}} \left(\left(i-\frac{41 \sqrt{31}}{403}\right) \sqrt{3}-\frac{123 i \sqrt{31}}{403}+1\right) \left(29+3 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}-\frac{16}{13}-\frac{4 \left(\left(i+\frac{5 \sqrt{31}}{62}\right) \sqrt{3}-\frac{15 i \sqrt{31}}{62}-1\right) 2^{\frac{2}{3}} \left(29+3 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{13}\right) \left(\left(\frac{199 \,2^{\frac{1}{3}} \left(\left(i+\frac{157 \sqrt{31}}{6169}\right) \sqrt{3}-\frac{1893 i \sqrt{31}}{6169}-\frac{53}{199}\right) \left(29+3 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}}{73}-\frac{46 \,2^{\frac{2}{3}} \left(\left(i-\frac{94 \sqrt{31}}{713}\right) \sqrt{3}-\frac{156 i \sqrt{31}}{713}+\frac{31}{23}\right) \left(29+3 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{73}+\frac{164}{73}+\frac{132 i \sqrt{31}}{2263}\right) \left(\frac{29 \left(\left(i+\frac{3 \sqrt{31}}{29}\right) \sqrt{3}-\frac{9 i \sqrt{31}}{29}-1\right) 2^{\frac{1}{3}} \left(29+3 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}}{24}-\frac{i \sqrt{3}\, \left(116+12 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}-\frac{\left(116+12 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}-\frac{2}{3}\right)^{-n}+\left(\left(\left(i-\frac{1025 \sqrt{31}}{2263}\right) \sqrt{3}-\frac{711 i \sqrt{31}}{2263}+\frac{325}{73}\right) 2^{\frac{1}{3}} \left(29+3 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}+\frac{8 \left(\left(i-\frac{125 \sqrt{31}}{124}\right) \sqrt{3}-\frac{63 i \sqrt{31}}{124}+\frac{25}{2}\right) 2^{\frac{2}{3}} \left(29+3 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{73}+\frac{164}{73}-\frac{132 i \sqrt{31}}{2263}\right) \left(\frac{\left(-3 \sqrt{31}\, \sqrt{3}+29\right) 2^{\frac{1}{3}} \left(29+3 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}}{12}+\frac{\left(116+12 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{2}{3}\right)^{-n}+\frac{136 \,2^{\frac{1}{3}} \cos \left(\frac{n \pi}{2}\right) \left(\left(i-\frac{7 \sqrt{31}}{68}\right) \sqrt{3}-\frac{21 i \sqrt{31}}{68}+1\right) \left(29+3 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}}{73}-\frac{19 \left(\left(i+\frac{\sqrt{31}}{19}\right) \sqrt{3}-\frac{3 i \sqrt{31}}{19}-1\right) \cos \left(\frac{n \pi}{2}\right) 2^{\frac{2}{3}} \left(29+3 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{73}+\frac{200 \cos \left(\frac{n \pi}{2}\right)}{73}+\frac{72 \left(-\frac{29 \left(\left(i-\frac{3 \sqrt{31}}{29}\right) \sqrt{3}-\frac{9 i \sqrt{31}}{29}+1\right) 2^{\frac{1}{3}} \left(29+3 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}}{24}+\frac{i \sqrt{3}\, \left(116+12 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}-\frac{\left(116+12 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}-\frac{2}{3}\right)^{-n}}{73}\right)}{5184}\)
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:
\(\begin{align*} F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{1}\! \left(x \right) &= 1\\ F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{4}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{4}\! \left(x \right) &= x\\ F_{5}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{4}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{13}\! \left(x \right)+F_{25}\! \left(x \right)\\ F_{12}\! \left(x \right) &= 0\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{19}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{4}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{22}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{2}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{20}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{13}\! \left(x \right)+F_{21}\! \left(x \right)\\ \end{align*}\)