Av(1342, 2143, 3124)
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Generating Function
\(\displaystyle -\frac{2 x^{4}+2 x^{3}-7 x^{2}+5 x -1}{\left(2 x^{2}-4 x +1\right) \left(x^{3}+x^{2}-2 x +1\right)}\)
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Counting Sequence
1, 1, 2, 6, 21, 74, 257, 883, 3019, 10306, 35174, 120063, 409878, 1399367, 4777721, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(2 x^{2}-4 x +1\right) \left(x^{3}+x^{2}-2 x +1\right) F \! \left(x \right)+2 x^{4}+2 x^{3}-7 x^{2}+5 x -1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(n +5\right) = -2 a \! \left(n \right)+2 a \! \left(n +1\right)+7 a \! \left(n +2\right)-11 a \! \left(n +3\right)+6 a \! \left(n +4\right), \quad n \geq 5\)
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Explicit Closed Form
\(\displaystyle -\frac{275 \left(\frac{2^{\frac{1}{3}} \left(\left(\mathrm{I}+\frac{27 \sqrt{31}}{31}\right) \sqrt{3}-\frac{81 \,\mathrm{I} \sqrt{31}}{31}-1\right) \left(47+3 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}}{14}+\frac{196}{5}+\left(\left(\mathrm{I}-\frac{6 \sqrt{31}}{31}\right) \sqrt{3}-\frac{18 \,\mathrm{I} \sqrt{31}}{31}+1\right) 2^{\frac{2}{3}} \left(47+3 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}\right) \left(\left(-\frac{31 \,2^{\frac{1}{3}} \left(\left(\mathrm{I}+\frac{485 \sqrt{31}}{961}\right) \sqrt{3}+\frac{919 \,\mathrm{I} \sqrt{31}}{961}-\frac{145}{31}\right) \left(47+3 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}}{462}+\frac{43 \,2^{\frac{2}{3}} \left(\left(\mathrm{I}+\frac{95 \sqrt{31}}{1333}\right) \sqrt{3}-\frac{529 \,\mathrm{I} \sqrt{31}}{1333}+\frac{25}{43}\right) \left(47+3 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{66}+\frac{350 \,\mathrm{I} \sqrt{31}}{1023}-\frac{1582}{165}\right) \left(\frac{47 \,2^{\frac{1}{3}} \left(\left(\mathrm{I}-\frac{3 \sqrt{31}}{47}\right) \sqrt{3}-\frac{9 \,\mathrm{I} \sqrt{31}}{47}+1\right) \left(47+3 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}}{1176}-\frac{\mathrm{I} \sqrt{3}\, \left(188+12 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{\left(188+12 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}-\frac{1}{3}\right)^{-n}+\left(-\frac{4 \,2^{\frac{1}{3}} \left(\left(\mathrm{I}-\frac{351 \sqrt{31}}{1364}\right) \sqrt{3}-\frac{67 \,\mathrm{I} \sqrt{31}}{682}+\frac{13}{44}\right) \left(47+3 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}}{21}+\frac{17 \,2^{\frac{2}{3}} \left(\left(\mathrm{I}-\frac{156 \sqrt{31}}{527}\right) \sqrt{3}-\frac{61 \,\mathrm{I} \sqrt{31}}{527}+\frac{26}{17}\right) \left(47+3 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{33}-\frac{350 \,\mathrm{I} \sqrt{31}}{1023}-\frac{1582}{165}\right) \left(\frac{2^{\frac{1}{3}} \left(3 \sqrt{31}\, \sqrt{3}-47\right) \left(47+3 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}}{588}-\frac{\left(188+12 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{1}{3}\right)^{-n}+\left(-\frac{68 \left(2^{\frac{1}{3}}-\frac{3 \,2^{\frac{5}{6}}}{8}\right) \left(\left(\mathrm{I}-\frac{\sqrt{31}}{17}\right) \sqrt{3}+\frac{3 \,\mathrm{I} \sqrt{31}}{17}-1\right) \left(47+3 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}}{231}-\left(\left(\mathrm{I}-\frac{\sqrt{31}}{11}\right) \sqrt{3}-\frac{3 \,\mathrm{I} \sqrt{31}}{11}+1\right) \left(2^{\frac{1}{6}}-\frac{4 \,2^{\frac{2}{3}}}{3}\right) \left(47+3 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}-\frac{5824}{165}+\frac{728 \sqrt{2}}{55}\right) \left(1-\frac{\sqrt{2}}{2}\right)^{-n}+\left(-\frac{68 \left(\left(\mathrm{I}-\frac{\sqrt{31}}{17}\right) \sqrt{3}+\frac{3 \,\mathrm{I} \sqrt{31}}{17}-1\right) \left(2^{\frac{1}{3}}+\frac{3 \,2^{\frac{5}{6}}}{8}\right) \left(47+3 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}}{231}+\left(2^{\frac{1}{6}}+\frac{4 \,2^{\frac{2}{3}}}{3}\right) \left(\left(\mathrm{I}-\frac{\sqrt{31}}{11}\right) \sqrt{3}-\frac{3 \,\mathrm{I} \sqrt{31}}{11}+1\right) \left(47+3 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}-\frac{5824}{165}-\frac{728 \sqrt{2}}{55}\right) \left(1+\frac{\sqrt{2}}{2}\right)^{-n}-\frac{644 \left(-\frac{47 \left(\left(\mathrm{I}+\frac{3 \sqrt{31}}{47}\right) \sqrt{3}-\frac{9 \,\mathrm{I} \sqrt{31}}{47}-1\right) 2^{\frac{1}{3}} \left(47+3 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}}{1176}+\frac{\mathrm{I} \sqrt{3}\, \left(188+12 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{\left(188+12 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}-\frac{1}{3}\right)^{-n}}{55}\right)}{1244208}\)
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This specification was found using the strategy pack "Point Placements" and has 43 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_{19}\! \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_{10}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\ 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_{18}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{15}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{12}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{20}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{22}\! \left(x \right)+F_{35}\! \left(x \right)\\ F_{21}\! \left(x \right) &= 0\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{25}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{7}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{22}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{34}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{20}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{36}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{37}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{38}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{4}\! \left(x \right) F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{42}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{29}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{39}\! \left(x \right)\\ \end{align*}\)