Av(1243, 1342, 2413, 2431, 3214)
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Generating Function
\(\displaystyle -\frac{x^{5}-2 x^{4}+3 x^{3}-6 x^{2}+4 x -1}{\left(x -1\right) \left(3 x^{3}-5 x^{2}+4 x -1\right)}\)
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Counting Sequence
1, 1, 2, 6, 19, 53, 136, 337, 828, 2036, 5016, 12369, 30505, 75224, 185479, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(x -1\right) \left(3 x^{3}-5 x^{2}+4 x -1\right) F \! \left(x \right)+x^{5}-2 x^{4}+3 x^{3}-6 x^{2}+4 x -1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 53\)
\(\displaystyle a \! \left(n +3\right) = 3 a \! \left(n \right)-5 a \! \left(n +1\right)+4 a \! \left(n +2\right)+1, \quad n \geq 6\)
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Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0 \\ 1 & n =1 \\ \frac{\left(248 \left(\left(\frac{123 \sqrt{31}}{124}+\mathrm{I}\right) \sqrt{3}+\frac{369 \,\mathrm{I} \sqrt{31}}{124}+1\right) 2^{\frac{1}{3}} \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}+600160+9889 \left(\left(\frac{57 \sqrt{31}}{899}+\mathrm{I}\right) \sqrt{3}-\frac{171 \,\mathrm{I} \sqrt{31}}{899}-1\right) 2^{\frac{2}{3}} \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{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}}{810216}\\+\\\frac{\left(-9889 \,2^{\frac{2}{3}} \left(\left(-\frac{57 \sqrt{31}}{899}+\mathrm{I}\right) \sqrt{3}-\frac{171 \,\mathrm{I} \sqrt{31}}{899}+1\right) \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}+600160-248 \left(\left(-\frac{123 \sqrt{31}}{124}+\mathrm{I}\right) \sqrt{3}+\frac{369 \,\mathrm{I} \sqrt{31}}{124}-1\right) 2^{\frac{1}{3}} \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}}{810216}\\-1+\\\frac{\left(\left(-1254 \,2^{\frac{2}{3}} \sqrt{31}\, \sqrt{3}+19778 \,2^{\frac{2}{3}}\right) \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}+600160+\left(-492 \sqrt{31}\, \sqrt{3}\, 2^{\frac{1}{3}}-496 \,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}}{810216} & \text{otherwise} \end{array}\right.\)
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This specification was found using the strategy pack "Point Placements" and has 54 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_{20}\! \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_{13}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{10}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \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) &= F_{11}\! \left(x \right)+F_{18}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{17}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{23}\! \left(x \right)+F_{42}\! \left(x \right)\\ F_{22}\! \left(x \right) &= 0\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{22}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{13}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{36}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{34}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{33}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{38}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{39}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{4}\! \left(x \right) F_{43}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{45}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{46}\! \left(x \right)+F_{53}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{4}\! \left(x \right) F_{47}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)+F_{51}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{49}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{22}\! \left(x \right)+F_{50}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{13}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{52}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{37}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{4}\! \left(x \right) F_{44}\! \left(x \right)\\ \end{align*}\)