Av(132, 2134, 2314, 3124)
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Generating Function
\(\displaystyle \frac{x -1}{x^{3}+2 x -1}\)
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Counting Sequence
1, 1, 2, 5, 11, 24, 53, 117, 258, 569, 1255, 2768, 6105, 13465, 29698, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(x^{3}+2 x -1\right) F \! \left(x \right)+1-x = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(n \right) = -2 a \! \left(n +2\right)+a \! \left(n +3\right), \quad n \geq 3\)
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Explicit Closed Form
\(\displaystyle -\frac{\left(\frac{\left(3 i \sqrt{59}-9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{576}-\frac{\left(i-\frac{\sqrt{59}}{9}\right) 3^{\frac{5}{6}} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{64}+\frac{i \sqrt{3}}{12}-\frac{1}{12}\right)^{-n} \left(\frac{3 \left(\left(i 3^{\frac{2}{3}}+3^{\frac{1}{6}}\right) \sqrt{59}-\frac{59 i 3^{\frac{1}{6}}}{3}-\frac{59 \,3^{\frac{2}{3}}}{9}\right) 2^{\frac{1}{3}} \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{8}+236+\sqrt{59}\, \left(i 3^{\frac{1}{3}}-\frac{3^{\frac{5}{6}}}{3}\right) 2^{\frac{2}{3}} \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}\right) \left(\frac{12 \left(-8 \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-8 i \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}+\left(\left(-i \sqrt{59}-3\right) 18^{\frac{1}{3}}+9 i 2^{\frac{1}{3}} 3^{\frac{1}{6}}+3^{\frac{1}{6}} 2^{\frac{1}{3}} \sqrt{59}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}\right)^{n} \left(i \sqrt{59}-767\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{-\frac{2 n}{3}} \left(\frac{\left(-3 i \sqrt{59}+9\right) \left(81+9 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+27 \,3^{\frac{1}{6}} \left(i-\frac{\sqrt{59}}{9}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-24 i 2^{\frac{1}{3}} 3^{\frac{5}{6}}+24 \,6^{\frac{1}{3}}}{3^{\frac{1}{6}} \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} \sqrt{59}-3 \left(81+9 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-8 \,6^{\frac{1}{3}}}\right)^{n}}{59}+\left(-24 \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+24 i \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}+\left(\left(3 i \sqrt{59}-9\right) 18^{\frac{1}{3}}-27 i 2^{\frac{1}{3}} 3^{\frac{1}{6}}+3 \,3^{\frac{1}{6}} 2^{\frac{1}{3}} \sqrt{59}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}\right)^{n} \left(\frac{61 \left(i 3^{\frac{2}{3}} \sqrt{59}-\frac{885 i 3^{\frac{1}{6}}}{61}-\frac{1121 \,3^{\frac{2}{3}}}{61}\right) 2^{\frac{1}{3}} \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}}{236}+\left(\left(\frac{21 i 3^{\frac{1}{3}}}{59}-\frac{38 \,3^{\frac{5}{6}}}{59}\right) \sqrt{59}+i 3^{\frac{5}{6}}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 2^{\frac{2}{3}}\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{-\frac{2 n}{3}} \left(\frac{\left(i \sqrt{59}+3\right) \left(81+9 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-9 \,3^{\frac{1}{6}} \left(i+\frac{\sqrt{59}}{9}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+8 i 2^{\frac{1}{3}} 3^{\frac{5}{6}}+8 \,6^{\frac{1}{3}}}{3^{\frac{1}{6}} \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} \sqrt{59}-3 \left(81+9 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-8 \,6^{\frac{1}{3}}}\right)^{n}+\frac{171 \left(2^{\frac{1}{3}} 3^{\frac{1}{6}} \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}} \left(\frac{\left(-i \sqrt{59}+3\right) \left(81+9 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+9 \,3^{\frac{1}{6}} \left(i-\frac{\sqrt{59}}{9}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-8 i 2^{\frac{1}{3}} 3^{\frac{5}{6}}+8 \,6^{\frac{1}{3}}}{2 \,3^{\frac{1}{6}} \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} \sqrt{59}-6 \left(81+9 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-16 \,6^{\frac{1}{3}}}\right)^{n} \sqrt{59}-\frac{3776}{19}\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{-\frac{2 n}{3}} \left(-48 \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-48 i \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}+\left(\left(-6 i \sqrt{59}-18\right) 18^{\frac{1}{3}}+54 i 2^{\frac{1}{3}} 3^{\frac{1}{6}}+6 \,3^{\frac{1}{6}} 2^{\frac{1}{3}} \sqrt{59}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}\right)^{n}}{236}+\frac{29 \left(\frac{59 \,3^{n +\frac{1}{2}} \left(i+\frac{17 \sqrt{59}}{59}\right) 2^{n} \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{-\frac{2 n}{3}+\frac{1}{3}}}{58}-\frac{177 \left(i 2^{n}-\frac{2^{n +1} \sqrt{59}}{708}\right) 3^{n +\frac{1}{2}} \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{-\frac{2 n}{3}+\frac{2}{3}}}{58}+\left(3^{n +\frac{2}{3}} \left(i \sqrt{59}+\frac{59}{58}\right) 2^{n +\frac{1}{3}} \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}+\frac{135 \,2^{n +\frac{2}{3}} \left(i \sqrt{59}-\frac{59}{45}\right) 3^{n +\frac{1}{3}} \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{58}-\frac{12 \,6^{n} \left(i \sqrt{59}+767\right)}{29}\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{-\frac{2 n}{3}}\right) \left(-8 \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+8 i \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{5}{6}}+\left(\left(i \sqrt{59}-3\right) 18^{\frac{1}{3}}-9 i 2^{\frac{1}{3}} 3^{\frac{1}{6}}+3^{\frac{1}{6}} 2^{\frac{1}{3}} \sqrt{59}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}\right)^{n}}{59}\right) \left(\left(-3 i \sqrt{59}-9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}+9 \left(i+\frac{\sqrt{59}}{9}\right) 3^{\frac{5}{6}} \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}-48 i \sqrt{3}-48\right)^{-n}}{101952}\)
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This specification was found using the strategy pack "Point Placements" and has 21 rules.

Found on January 18, 2022.

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\(\begin{align*} F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{1}\! \left(x \right) &= 1\\ F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{4}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{4}\! \left(x \right) &= x\\ F_{5}\! \left(x \right) &= F_{6}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{4}\! \left(x \right) F_{6}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{12}\! \left(x \right)+F_{18}\! \left(x \right)\\ F_{11}\! \left(x \right) &= 0\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{15}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{4}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{12}\! \left(x \right)+F_{17}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{2}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{20}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{18}\! \left(x \right)\\ \end{align*}\)