Av(1342, 2314, 2413, 4123)
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Generating Function
\(\displaystyle -\frac{\left(2 x -1\right) \left(4 x^{6}-14 x^{5}+27 x^{4}-30 x^{3}+20 x^{2}-7 x +1\right)}{\left(x^{2}-3 x +1\right) \left(3 x^{3}-5 x^{2}+4 x -1\right) \left(x -1\right)^{3}}\)
Counting Sequence
1, 1, 2, 6, 20, 64, 194, 566, 1612, 4521, 12548, 34564, 94654, 257995, 700455, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{2}-3 x +1\right) \left(3 x^{3}-5 x^{2}+4 x -1\right) \left(x -1\right)^{3} F \! \left(x \right)+\left(2 x -1\right) \left(4 x^{6}-14 x^{5}+27 x^{4}-30 x^{3}+20 x^{2}-7 x +1\right) = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 64\)
\(\displaystyle a \! \left(6\right) = 194\)
\(\displaystyle a \! \left(7\right) = 566\)
\(\displaystyle a \! \left(n +5\right) = -\frac{n^{2}}{2}+3 a \! \left(n \right)-14 a \! \left(n +1\right)+22 a \! \left(n +2\right)-18 a \! \left(n +3\right)+7 a \! \left(n +4\right)+\frac{n}{2}-1, \quad n \geq 8\)
Explicit Closed Form
\(\displaystyle \frac{\left(3410 \left(\left(-\frac{48 \,\mathrm{I}}{31}+\frac{16 \sqrt{3}}{31}\right) \sqrt{31}+\mathrm{I} \sqrt{3}-1\right) 2^{\frac{2}{3}} \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}-150040+5425 \,2^{\frac{1}{3}} \left(\left(-\frac{129 \,\mathrm{I}}{1085}-\frac{43 \sqrt{3}}{1085}\right) \sqrt{31}+\mathrm{I} \sqrt{3}+1\right) \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{47 \,2^{\frac{1}{3}} \left(\left(\mathrm{I}-\frac{9 \sqrt{31}}{47}\right) \sqrt{3}-\frac{27 \,\mathrm{I} \sqrt{31}}{47}+1\right) \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}}{8712}-\frac{\mathrm{I} \sqrt{3}\, \left(188+36 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{36}+\frac{\left(188+36 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{36}+\frac{5}{9}\right)^{-n}}{450120}+\frac{\left(-3410 \left(\left(-\frac{48 \,\mathrm{I}}{31}-\frac{16 \sqrt{3}}{31}\right) \sqrt{31}+\mathrm{I} \sqrt{3}+1\right) 2^{\frac{2}{3}} \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}-150040-5425 \,2^{\frac{1}{3}} \left(\left(-\frac{129 \,\mathrm{I}}{1085}+\frac{43 \sqrt{3}}{1085}\right) \sqrt{31}+\mathrm{I} \sqrt{3}-1\right) \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(-\frac{47 \left(\left(\mathrm{I}+\frac{9 \sqrt{31}}{47}\right) \sqrt{3}-\frac{27 \,\mathrm{I} \sqrt{31}}{47}-1\right) 2^{\frac{1}{3}} \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}}{8712}+\frac{\mathrm{I} \sqrt{3}\, \left(188+36 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{36}+\frac{\left(188+36 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{36}+\frac{5}{9}\right)^{-n}}{450120}+\frac{\left(\left(-3520 \sqrt{31}\, 2^{\frac{2}{3}} \sqrt{3}+6820 \,2^{\frac{2}{3}}\right) \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}-150040+\left(430 \sqrt{31}\, \sqrt{3}\, 2^{\frac{1}{3}}-10850 \,2^{\frac{1}{3}}\right) \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{2^{\frac{1}{3}} \left(9 \sqrt{31}\, \sqrt{3}-47\right) \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}}{4356}-\frac{\left(188+36 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{18}+\frac{5}{9}\right)^{-n}}{450120}+\frac{\left(-90024 \sqrt{5}+900240\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{450120}+\frac{\left(90024 \sqrt{5}+900240\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{450120}-\frac{n^{2}}{2}+\frac{3 n}{2}-2\)

This specification was found using the strategy pack "Point Placements" and has 55 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_{13}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{13}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{5}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{14}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right) F_{13}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x \right)\\ F_{13}\! \left(x \right) &= x\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{18}\! \left(x \right)\\ F_{15}\! \left(x \right) &= 0\\ F_{16}\! \left(x \right) &= F_{13}\! \left(x \right) F_{17}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{13}\! \left(x \right) F_{19}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{24}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{13}\! \left(x \right) F_{23}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{13}\! \left(x \right) F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{25}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{13}\! \left(x \right) F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{33}\! \left(x \right)+F_{37}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{13}\! \left(x \right) F_{34}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{36}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{6}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{32}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{13}\! \left(x \right) F_{38}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{44}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{13}\! \left(x \right) F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{43}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{40}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{52}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{46}\! \left(x \right)+F_{50}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{13}\! \left(x \right) F_{47}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)+F_{49}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{10}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{45}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{13}\! \left(x \right) F_{51}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{45}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{13}\! \left(x \right) F_{54}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{52}\! \left(x \right)\\ \end{align*}\)