Av(123, 1432, 2143, 2413)
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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
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(n \right) = -2 a \! \left(2+n \right)+a \! \left(n +3\right), \quad n \geq 3\)
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Explicit Closed Form
\(\displaystyle -\frac{61 \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} \left(\left(\frac{84 \,2^{\frac{2}{3}-n} \left(\left(\frac{59 i}{21}-\frac{38 \sqrt{59}}{21}\right) 3^{\frac{5}{6}+\frac{n}{3}}+i \sqrt{59}\, 3^{\frac{1}{3}+\frac{n}{3}}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}-\frac{n}{3}}}{61}+2^{\frac{1}{3}-n} \left(\left(i \sqrt{59}-\frac{1121}{61}\right) 3^{\frac{2}{3}+\frac{n}{3}}-\frac{885 i 3^{\frac{1}{6}+\frac{n}{3}}}{61}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}-\frac{n}{3}}\right) \left(\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}}\right)^{n} \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}}}{\sqrt{59}\, \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{1}{6}}-8 \,6^{\frac{1}{3}}-3 \left(81+9 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}\right)^{n}+\frac{48 \left(18+2 \sqrt{59}\, \sqrt{3}\right)^{\frac{n}{3}} 3^{n} \left(i \sqrt{59}-767\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{-\frac{2 n}{3}} \left(\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}}\right)^{n} \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}}}{\sqrt{59}\, \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{1}{6}}-8 \,6^{\frac{1}{3}}-3 \left(81+9 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}\right)^{n}}{61}+\left(\frac{118 \left(\frac{17 \sqrt{59}}{59}+i\right) 2^{n} 3^{n +\frac{1}{2}} \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{-\frac{2 n}{3}+\frac{1}{3}}}{61}-\frac{354 \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}}}{61}+\frac{116 \left(2^{n +\frac{1}{3}} 3^{n +\frac{2}{3}} \left(i \sqrt{59}+\frac{59}{58}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}+\frac{135 \,3^{n +\frac{1}{3}} \left(i \sqrt{59}-\frac{59}{45}\right) 2^{n +\frac{2}{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}}}{61}\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}}+2^{\frac{1}{3}} 3^{\frac{1}{6}} \sqrt{59}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}\right)^{n}+\frac{171 \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{-\frac{2 n}{3}} \left(\sqrt{59}\, 3^{\frac{1}{6}+n} 2^{\frac{1}{3}} \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}} \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}}+2^{\frac{1}{3}} 3^{\frac{1}{6}} \sqrt{59}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}\right)^{n} \left(-\frac{1}{\sqrt{59}\, \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}} 3^{\frac{1}{6}}-8 \,6^{\frac{1}{3}}-3 \left(81+9 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}\right)^{n} \left(\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}}\right)^{n}-\frac{3776 \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 \,2^{\frac{1}{3}} 3^{\frac{1}{6}} \sqrt{59}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{2}{3}}\right)^{n}}{19}\right)}{61}\right) \left(\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}}+\frac{1888}{3}+\frac{8 \,2^{\frac{2}{3}} \sqrt{59}\, \left(i 3^{\frac{1}{3}}-\frac{3^{\frac{5}{6}}}{3}\right) \left(9+\sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{3}\right) \left(\frac{\left(3 i \sqrt{59}-9\right) \left(108+12 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{576}-\frac{3^{\frac{5}{6}} \left(i-\frac{\sqrt{59}}{9}\right) \left(36+4 \sqrt{59}\, \sqrt{3}\right)^{\frac{1}{3}}}{64}+\frac{i \sqrt{3}}{12}-\frac{1}{12}\right)^{-n}}{64161792}\)
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This specification was found using the strategy pack "Point Placements" and has 26 rules.

Found on January 18, 2022.

Finding the specification took 1 seconds.

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