Av(1432, 2341, 2413, 3142)
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Generating Function
\(\displaystyle \frac{2 x^{3}+x^{2}+2 x -1}{2 x^{3}+3 x -1}\)
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Counting Sequence
1, 1, 2, 6, 20, 64, 204, 652, 2084, 6660, 21284, 68020, 217380, 694708, 2220164, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(2 x^{3}+3 x -1\right) F \! \left(x \right)-2 x^{3}-x^{2}-2 x +1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(n \right) = -\frac{3 a \! \left(n +2\right)}{2}+\frac{a \! \left(n +3\right)}{2}, \quad n \geq 4\)
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Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0 \\ -\frac{\left(2+2 \sqrt{3}\right)^{-\frac{2 n}{3}+\frac{2}{3}} \sqrt{3}\, \left(\frac{\left(-1+3 \,\mathrm{I}+\left(1-\mathrm{I}\right) \sqrt{3}\right) \left(2+2 \sqrt{3}\right)^{\frac{1}{3}}}{32}+\frac{\mathrm{I} \sqrt{3}}{16}-\frac{1}{16}\right)^{-n} \left(\frac{\left(1-3 \,\mathrm{I}+\left(-1+\mathrm{I}\right) \sqrt{3}\right) \left(2+2 \sqrt{3}\right)^{\frac{1}{3}}-2 \,\mathrm{I} \sqrt{3}+2}{\left(\sqrt{3}-1\right) \left(2+2 \sqrt{3}\right)^{\frac{1}{3}}-2}\right)^{n} \left(\left(-2 \,\mathrm{I} \sqrt{3}-2\right) \left(2+2 \sqrt{3}\right)^{\frac{1}{3}}+\left(-1-3 \,\mathrm{I}+\left(1+\mathrm{I}\right) \sqrt{3}\right) 2^{\frac{2}{3}} \left(1+\sqrt{3}\right)^{\frac{2}{3}}\right)^{n} \left(\left(-8-24 \,\mathrm{I}+\left(8+8 \,\mathrm{I}\right) \sqrt{3}\right) \left(2+2 \sqrt{3}\right)^{\frac{1}{3}}-16 \,\mathrm{I} \sqrt{3}-16\right)^{-n}}{18}\\+\\\frac{\sqrt{3}\, \left(\left(\sqrt{3}-1\right) \left(2+2 \sqrt{3}\right)^{\frac{1}{3}}-2\right)^{-n} \left(\left(1-3 \,\mathrm{I}+\left(-1+\mathrm{I}\right) \sqrt{3}\right) \left(2+2 \sqrt{3}\right)^{\frac{1}{3}}-2 \,\mathrm{I} \sqrt{3}+2\right)^{-n} \left(\frac{\left(-1-3 \,\mathrm{I}+\left(1+\mathrm{I}\right) \sqrt{3}\right) \left(2+2 \sqrt{3}\right)^{\frac{1}{3}}}{64}-\frac{\mathrm{I} \sqrt{3}}{32}-\frac{1}{32}\right)^{-n} 2^{\frac{1}{3}} \left(\frac{\left(-2 \,\mathrm{I} \sqrt{3}-2\right) \left(2+2 \sqrt{3}\right)^{\frac{1}{3}}}{2}+\frac{\left(-1-3 \,\mathrm{I}+\left(1+\mathrm{I}\right) \sqrt{3}\right) 2^{\frac{2}{3}} \left(1+\sqrt{3}\right)^{\frac{2}{3}}}{2}\right)^{n} \left(\frac{\left(2 \,\mathrm{I} \sqrt{3}-2\right) \left(2+2 \sqrt{3}\right)^{\frac{1}{3}}}{16}-\frac{2^{\frac{2}{3}} \left(1+\sqrt{3}\right)^{\frac{2}{3}} \left(1-3 \,\mathrm{I}+\left(-1+\mathrm{I}\right) \sqrt{3}\right)}{16}\right)^{n} \left(1+\sqrt{3}\right)^{-n +\frac{1}{3}}}{36}\\+\\\frac{\left(\left(\sqrt{3}-1\right) \left(2+2 \sqrt{3}\right)^{\frac{1}{3}}-2\right)^{-n} \left(\left(1-3 \,\mathrm{I}+\left(-1+\mathrm{I}\right) \sqrt{3}\right) \left(2+2 \sqrt{3}\right)^{\frac{1}{3}}-2 \,\mathrm{I} \sqrt{3}+2\right)^{-n} \left(\frac{\left(-1-3 \,\mathrm{I}+\left(1+\mathrm{I}\right) \sqrt{3}\right) \left(2+2 \sqrt{3}\right)^{\frac{1}{3}}}{256}-\frac{\mathrm{I} \sqrt{3}}{128}-\frac{1}{128}\right)^{-n} \left(\frac{\left(2 \,\mathrm{I} \sqrt{3}-2\right) \left(2+2 \sqrt{3}\right)^{\frac{1}{3}}}{8}-\frac{2^{\frac{2}{3}} \left(1+\sqrt{3}\right)^{\frac{2}{3}} \left(1-3 \,\mathrm{I}+\left(-1+\mathrm{I}\right) \sqrt{3}\right)}{8}\right)^{n} \left(2^{\frac{1}{3}} \left(1+\sqrt{3}\right)^{-n +\frac{1}{3}}+\left(1+\sqrt{3}\right)^{-n +\frac{2}{3}} 2^{\frac{2}{3}}\right) \left(\frac{\left(-2 \,\mathrm{I} \sqrt{3}-2\right) \left(2+2 \sqrt{3}\right)^{\frac{1}{3}}}{16}+\frac{\left(-1-3 \,\mathrm{I}+\left(1+\mathrm{I}\right) \sqrt{3}\right) 2^{\frac{2}{3}} \left(1+\sqrt{3}\right)^{\frac{2}{3}}}{16}\right)^{n}}{12}\\-\\\frac{\left(\frac{\left(-1+3 \,\mathrm{I}+\left(1-\mathrm{I}\right) \sqrt{3}\right) \left(2+2 \sqrt{3}\right)^{\frac{1}{3}}}{64}+\frac{\mathrm{I} \sqrt{3}}{32}-\frac{1}{32}\right)^{-n} \left(\left(-1-3 \,\mathrm{I}+\left(1+\mathrm{I}\right) \sqrt{3}\right) \left(2+2 \sqrt{3}\right)^{\frac{1}{3}}-2 \,\mathrm{I} \sqrt{3}-2\right)^{-n} \left(\left(\left(1-2 \,\mathrm{I}+\left(-\frac{2}{3}+\mathrm{I}\right) \sqrt{3}\right) \left(2+2 \sqrt{3}\right)^{-\frac{2 n}{3}+\frac{2}{3}}-\left(-1+\mathrm{I}+\left(-\frac{1}{3}+\mathrm{I}\right) \sqrt{3}\right) \left(2+2 \sqrt{3}\right)^{-\frac{2 n}{3}+\frac{1}{3}}\right) \left(\frac{\left(-2 \,\mathrm{I} \sqrt{3}-2\right) \left(2+2 \sqrt{3}\right)^{\frac{1}{3}}}{8}+\frac{\left(-1-3 \,\mathrm{I}+\left(1+\mathrm{I}\right) \sqrt{3}\right) 2^{\frac{2}{3}} \left(1+\sqrt{3}\right)^{\frac{2}{3}}}{8}\right)^{n}-\left(\frac{\left(2 \,\mathrm{I} \sqrt{3}-2\right) \left(2+2 \sqrt{3}\right)^{\frac{1}{3}}}{8}-\frac{2^{\frac{2}{3}} \left(1+\sqrt{3}\right)^{\frac{2}{3}} \left(1-3 \,\mathrm{I}+\left(-1+\mathrm{I}\right) \sqrt{3}\right)}{8}\right)^{n} \left(\left(-1-2 \,\mathrm{I}+\left(\frac{2}{3}+\mathrm{I}\right) \sqrt{3}\right) \left(2+2 \sqrt{3}\right)^{-\frac{2 n}{3}+\frac{2}{3}}-\left(2+2 \sqrt{3}\right)^{-\frac{2 n}{3}+\frac{1}{3}} \left(1+\mathrm{I}+\left(\frac{1}{3}+\mathrm{I}\right) \sqrt{3}\right)\right)\right)}{24} & \text{otherwise} \end{array}\right.\)
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This specification was found using the strategy pack "Point Placements" and has 64 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_{12}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{22}\! \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_{12}\! \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) &= x\\ F_{13}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{12}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{20}\! \left(x \right)\\ F_{15}\! \left(x \right) &= 0\\ F_{16}\! \left(x \right) &= F_{12}\! \left(x \right) F_{17}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{12}\! \left(x \right) F_{6}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{12}\! \left(x \right) F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{9}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{39}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{12}\! \left(x \right) F_{25}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{12}\! \left(x \right) F_{13}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{31}\! \left(x \right)+F_{35}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{12}\! \left(x \right) F_{32}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{34}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{27}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{30}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{12}\! \left(x \right) F_{36}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{37}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{12}\! \left(x \right) F_{23}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{40}\! \left(x \right)+F_{55}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{12}\! \left(x \right) F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{45}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{12}\! \left(x \right) F_{17}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{46}\! \left(x \right)\\ F_{46}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{47}\! \left(x \right)+F_{51}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{12}\! \left(x \right) F_{48}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)+F_{50}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{43}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{46}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{12}\! \left(x \right) F_{52}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{53}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{12}\! \left(x \right) F_{39}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{12}\! \left(x \right) F_{56}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{57}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)+F_{60}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{59}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{12}\! \left(x \right) F_{36}\! \left(x \right)\\ F_{60}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{61}\! \left(x \right)+F_{62}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{12}\! \left(x \right) F_{52}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{12}\! \left(x \right) F_{63}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{57}\! \left(x \right)\\ \end{align*}\)