Av(123, 2143, 2413)
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Generating Function
\(\displaystyle -\frac{\left(x -1\right)^{2}}{x^{3}-2 x^{2}+3 x -1}\)
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Counting Sequence
1, 1, 2, 5, 12, 28, 65, 151, 351, 816, 1897, 4410, 10252, 23833, 55405, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(x^{3}-2 x^{2}+3 x -1\right) F \! \left(x \right)+\left(x -1\right)^{2} = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(n +3\right) = a \! \left(n \right)-2 a \! \left(n +1\right)+3 a \! \left(n +2\right), \quad n \geq 3\)
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Explicit Closed Form
\(\displaystyle \frac{\left(\left(-\frac{3 \left(\left(i+\frac{5 \sqrt{23}}{207}\right) \sqrt{3}-\frac{133 i \sqrt{23}}{207}+\frac{5}{9}\right) 2^{\frac{1}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}}{200}+\frac{\left(\left(i-\frac{4 \sqrt{23}}{69}\right) \sqrt{3}+\frac{4 i \sqrt{23}}{69}+\frac{1}{3}\right) 2^{\frac{2}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{20}-\frac{i \sqrt{23}}{138}+\frac{5}{6}\right) \left(\frac{11 \left(\left(i-\frac{3 \sqrt{23}}{11}\right) \sqrt{3}-\frac{9 i \sqrt{23}}{11}+1\right) 2^{\frac{1}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}}{600}-\frac{i \sqrt{3}\, \left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{\left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{2}{3}\right)^{-n}+\left(-\frac{2^{\frac{1}{3}} \left(\left(i+\frac{32 \sqrt{23}}{23}\right) \sqrt{3}-\frac{37 i \sqrt{23}}{23}-8\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}}{300}+\frac{\left(\left(i+\frac{2 \sqrt{23}}{23}\right) \sqrt{3}-\frac{2 i \sqrt{23}}{23}+2\right) 2^{\frac{2}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{30}+\frac{i \sqrt{23}}{138}+\frac{5}{6}\right) \left(\frac{2^{\frac{1}{3}} \left(3 \sqrt{23}\, \sqrt{3}-11\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}}{300}-\frac{\left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}+\frac{2}{3}\right)^{-n}+\left(-\frac{11 \left(\left(i+\frac{3 \sqrt{23}}{11}\right) \sqrt{3}-\frac{9 i \sqrt{23}}{11}-1\right) 2^{\frac{1}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}}{600}+\frac{i \sqrt{3}\, \left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{\left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{2}{3}\right)^{-n}\right) \left(\left(\left(i+\frac{4 \sqrt{23}}{23}\right) \sqrt{3}-\frac{12 i \sqrt{23}}{23}-1\right) 2^{\frac{1}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}+100-\frac{5 \left(\left(i+\frac{\sqrt{23}}{23}\right) \sqrt{3}+\frac{3 i \sqrt{23}}{23}+1\right) 2^{\frac{2}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{2}\right)}{300}\)
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This specification was found using the strategy pack "Point Placements" and has 25 rules.

Found on January 18, 2022.

Finding the specification took 2 seconds.

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