Av(1342, 2143, 2314)
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Generating Function
\(\displaystyle \frac{\left(2 x -1\right) \left(x^{2}-3 x +1\right)}{7 x^{3}-11 x^{2}+6 x -1}\)
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Counting Sequence
1, 1, 2, 6, 21, 74, 255, 863, 2891, 9638, 32068, 106627, 354480, 1178459, 3917863, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(7 x^{3}-11 x^{2}+6 x -1\right) F \! \left(x \right)-\left(2 x -1\right) \left(x^{2}-3 x +1\right) = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(n +3\right) = 7 a \! \left(n \right)-11 a \! \left(n +1\right)+6 a \! \left(n +2\right), \quad n \geq 4\)
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Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(-2231 \left(\left(\mathrm{I}-\frac{287 \sqrt{23}}{2231}\right) \sqrt{3}-\frac{861 \,\mathrm{I} \sqrt{23}}{2231}+1\right) 2^{\frac{1}{3}} \left(173+21 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}+23000+2990 \,2^{\frac{2}{3}} \left(\left(\mathrm{I}+\frac{28 \sqrt{23}}{299}\right) \sqrt{3}-\frac{84 \,\mathrm{I} \sqrt{23}}{299}-1\right) \left(173+21 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}\right) \left(\frac{173 \left(\left(\mathrm{I}-\frac{21 \sqrt{23}}{173}\right) \sqrt{3}-\frac{63 \,\mathrm{I} \sqrt{23}}{173}+1\right) 2^{\frac{1}{3}} \left(173+21 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}}{4200}-\frac{\mathrm{I} \sqrt{3}\, \left(692+84 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{84}+\frac{\left(692+84 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{84}+\frac{11}{21}\right)^{-n}}{96600}\\+\\\frac{\left(-2990 \,2^{\frac{2}{3}} \left(\left(\mathrm{I}-\frac{28 \sqrt{23}}{299}\right) \sqrt{3}-\frac{84 \,\mathrm{I} \sqrt{23}}{299}+1\right) \left(173+21 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}+23000+2231 \left(\left(\mathrm{I}+\frac{287 \sqrt{23}}{2231}\right) \sqrt{3}-\frac{861 \,\mathrm{I} \sqrt{23}}{2231}-1\right) 2^{\frac{1}{3}} \left(173+21 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(-\frac{173 \left(\left(\mathrm{I}+\frac{21 \sqrt{23}}{173}\right) \sqrt{3}-\frac{63 \,\mathrm{I} \sqrt{23}}{173}-1\right) 2^{\frac{1}{3}} \left(173+21 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}}{4200}+\frac{\mathrm{I} \sqrt{3}\, \left(692+84 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{84}+\frac{\left(692+84 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{84}+\frac{11}{21}\right)^{-n}}{96600}\\-\\\frac{41 \left(\frac{40 \,2^{\frac{2}{3}} \left(\sqrt{23}\, \sqrt{3}-\frac{299}{28}\right) \left(173+21 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{41}-\frac{11500}{287}+2^{\frac{1}{3}} \left(\sqrt{23}\, \sqrt{3}-\frac{2231}{287}\right) \left(173+21 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{2^{\frac{1}{3}} \left(21 \sqrt{23}\, \sqrt{3}-173\right) \left(173+21 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}}{2100}-\frac{\left(692+84 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{42}+\frac{11}{21}\right)^{-n}}{6900} & \text{otherwise} \end{array}\right.\)
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This specification was found using the strategy pack "Point Placements" and has 29 rules.

Found on July 23, 2021.

Finding the specification took 4 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_{7}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{7}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{7}\! \left(x \right) &= x\\ F_{8}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{11}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{7}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{0}\! \left(x \right) F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x \right) F_{26}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{19}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{19}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{22}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{21}\! \left(x \right) F_{24}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{25}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{24}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{25}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{24} \left(x \right)^{2} F_{19}\! \left(x \right) F_{7}\! \left(x \right)\\ \end{align*}\)