Av(1324, 2413, 3241, 4132)
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Generating Function
\(\displaystyle \frac{x^{4}-x^{3}+7 x^{2}-5 x +1}{\left(x^{2}-3 x +1\right)^{2}}\)
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Counting Sequence
1, 1, 2, 6, 20, 65, 204, 623, 1864, 5490, 15970, 45991, 131352, 372541, 1050350, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(x^{2}-3 x +1\right)^{2} F \! \left(x \right)-x^{4}+x^{3}-7 x^{2}+5 x -1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(n +4\right) = -a \! \left(n \right)+6 a \! \left(n +1\right)-11 a \! \left(n +2\right)+6 a \! \left(n +3\right), \quad n \geq 5\)
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Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0 \\ -\frac{n \left(2 \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n} \sqrt{5}-2 \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n} \sqrt{5}-5 \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}-5 \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}\right)}{5} & \text{otherwise} \end{array}\right.\)
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This specification was found using the strategy pack "Point Placements" and has 33 rules.

Found on January 18, 2022.

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\(\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_{13}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{14}\! \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_{13}\! \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_{6}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right) F_{13}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{9}\! \left(x \right)\\ F_{13}\! \left(x \right) &= x\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{17}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{16}\! \left(x \right) &= 0\\ F_{17}\! \left(x \right) &= F_{13}\! \left(x \right) F_{18}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{13}\! \left(x \right) F_{22}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{19}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{24}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{13}\! \left(x \right) F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{23}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{13}\! \left(x \right) F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{13}\! \left(x \right) F_{32}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{29}\! \left(x \right)\\ \end{align*}\)