Av(1234, 2143, 2341, 2413)
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Generating Function
\(\displaystyle \frac{x^{6}-3 x^{5}+6 x^{4}-7 x^{3}+9 x^{2}-5 x +1}{\left(x^{3}-2 x^{2}+3 x -1\right)^{2}}\)
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Counting Sequence
1, 1, 2, 6, 20, 61, 174, 478, 1282, 3378, 8778, 22557, 57438, 145149, 364448, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(x^{3}-2 x^{2}+3 x -1\right)^{2} F \! \left(x \right)-x^{6}+3 x^{5}-6 x^{4}+7 x^{3}-9 x^{2}+5 x -1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 61\)
\(\displaystyle a \! \left(6\right) = 174\)
\(\displaystyle a \! \left(n +6\right) = -a \! \left(n \right)+4 a \! \left(n +1\right)-10 a \! \left(n +2\right)+14 a \! \left(n +3\right)-13 a \! \left(n +4\right)+6 a \! \left(n +5\right), \quad n \geq 7\)
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Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(207 \left(\left(\left(-\frac{7 n}{9}-\frac{170}{69}\right) \sqrt{23}+\mathrm{I} n +10 \,\mathrm{I}\right) \sqrt{3}+\left(-\frac{7 \,\mathrm{I} n}{3}-\frac{170 \,\mathrm{I}}{23}\right) \sqrt{23}+n +10\right) 2^{\frac{1}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}-3105 \left(\left(\left(\frac{n}{27}-\frac{4}{207}\right) \sqrt{23}+\mathrm{I} n +\frac{8 \,\mathrm{I}}{3}\right) \sqrt{3}+\left(-\frac{\mathrm{I} n}{9}+\frac{4 \,\mathrm{I}}{69}\right) \sqrt{23}-n -\frac{8}{3}\right) 2^{\frac{2}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}+55200 n \right) \left(\frac{11 \,2^{\frac{1}{3}} \left(\left(\mathrm{I}-\frac{3 \sqrt{23}}{11}\right) \sqrt{3}-\frac{9 \,\mathrm{I} \sqrt{23}}{11}+1\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}}{600}-\frac{\mathrm{I} \sqrt{3}\, \left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{\left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{2}{3}\right)^{-n}}{317400}\\+\\\frac{\left(3105 \left(\left(\left(-\frac{n}{27}+\frac{4}{207}\right) \sqrt{23}+\mathrm{I} n +\frac{8 \,\mathrm{I}}{3}\right) \sqrt{3}+\left(-\frac{\mathrm{I} n}{9}+\frac{4 \,\mathrm{I}}{69}\right) \sqrt{23}+n +\frac{8}{3}\right) 2^{\frac{2}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}-207 \left(\left(\left(\frac{7 n}{9}+\frac{170}{69}\right) \sqrt{23}+\mathrm{I} n +10 \,\mathrm{I}\right) \sqrt{3}+\left(-\frac{7 \,\mathrm{I} n}{3}-\frac{170 \,\mathrm{I}}{23}\right) \sqrt{23}-n -10\right) 2^{\frac{1}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}+55200 n \right) \left(-\frac{11 \,2^{\frac{1}{3}} \left(\left(\mathrm{I}+\frac{3 \sqrt{23}}{11}\right) \sqrt{3}-\frac{9 \,\mathrm{I} \sqrt{23}}{11}-1\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}}{600}+\frac{\mathrm{I} \sqrt{3}\, \left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{\left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{2}{3}\right)^{-n}}{317400}\\+\\\frac{7 \left(\frac{5 \left(\sqrt{3}\, \left(n -\frac{12}{23}\right) \sqrt{23}-27 n -72\right) 2^{\frac{2}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{7}+\left(\sqrt{3}\, \left(n +\frac{510}{161}\right) \sqrt{23}-\frac{9 n}{7}-\frac{90}{7}\right) 2^{\frac{1}{3}} \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}+\frac{1200 n}{7}\right) \left(\frac{2^{\frac{1}{3}} \left(3 \sqrt{23}\, \sqrt{3}-11\right) \left(11+3 \sqrt{23}\, \sqrt{3}\right)^{\frac{2}{3}}}{300}-\frac{\left(44+12 \sqrt{23}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}+\frac{2}{3}\right)^{-n}}{6900} & \text{otherwise} \end{array}\right.\)
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This specification was found using the strategy pack "Point Placements" and has 110 rules.

Found on January 18, 2022.

Finding the specification took 3 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_{29}\! \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_{16}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{15}\! \left(x \right) &= 0\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{21}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{11}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{24}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{4}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{26}\! \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_{26}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{104}\! \left(x \right)+F_{15}\! \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_{45}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{37}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{36}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{38}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{39}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{4}\! \left(x \right) F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{44}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{4}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{40}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{80}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{31}\! \left(x \right)+F_{47}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{4}\! \left(x \right) F_{48}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{49}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{4}\! \left(x \right) F_{51}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)+F_{53}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{40}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{49}\! \left(x \right)+F_{54}\! \left(x \right)\\ F_{54}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{55}\! \left(x \right)+F_{76}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{4}\! \left(x \right) F_{56}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)+F_{64}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)+F_{61}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{59}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{4}\! \left(x \right) F_{60}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{4}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{62}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{4}\! \left(x \right) F_{63}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{40}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{65}\! \left(x \right)+F_{68}\! \left(x \right)\\ F_{65}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{55}\! \left(x \right)+F_{66}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{4}\! \left(x \right) F_{67}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{49}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{69}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{4}\! \left(x \right) F_{70}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{71}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{72}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{4}\! \left(x \right) F_{73}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{74}\! \left(x \right)+F_{75}\! \left(x \right)\\ F_{74}\! \left(x \right) &= F_{49}\! \left(x \right)\\ F_{75}\! \left(x \right) &= F_{71}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{4}\! \left(x \right) F_{77}\! \left(x \right)\\ F_{77}\! \left(x \right) &= F_{78}\! \left(x \right)+F_{79}\! \left(x \right)\\ F_{78}\! \left(x \right) &= F_{49}\! \left(x \right)\\ F_{79}\! \left(x \right) &= F_{71}\! \left(x \right)\\ F_{80}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{81}\! \left(x \right)+F_{89}\! \left(x \right)+F_{97}\! \left(x \right)\\ F_{81}\! \left(x \right) &= F_{4}\! \left(x \right) F_{82}\! \left(x \right)\\ F_{82}\! \left(x \right) &= F_{83}\! \left(x \right)+F_{86}\! \left(x \right)\\ F_{83}\! \left(x \right) &= F_{84}\! \left(x \right)+F_{85}\! \left(x \right)\\ F_{84}\! \left(x \right) &= F_{59}\! \left(x \right)\\ F_{85}\! \left(x \right) &= F_{62}\! \left(x \right)\\ F_{86}\! \left(x \right) &= F_{87}\! \left(x \right)+F_{88}\! \left(x \right)\\ F_{87}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{66}\! \left(x \right)+F_{81}\! \left(x \right)\\ F_{88}\! \left(x \right) &= F_{69}\! \left(x \right)\\ F_{89}\! \left(x \right) &= F_{4}\! \left(x \right) F_{90}\! \left(x \right)\\ F_{90}\! \left(x \right) &= F_{91}\! \left(x \right)+F_{92}\! \left(x \right)\\ F_{91}\! \left(x \right) &= F_{49}\! \left(x \right)\\ F_{92}\! \left(x \right) &= F_{93}\! \left(x \right)\\ F_{93}\! \left(x \right) &= F_{4}\! \left(x \right) F_{94}\! \left(x \right)\\ F_{94}\! \left(x \right) &= F_{95}\! \left(x \right)+F_{96}\! \left(x \right)\\ F_{95}\! \left(x \right) &= F_{49}\! \left(x \right)\\ F_{96}\! \left(x \right) &= F_{93}\! \left(x \right)\\ F_{97}\! \left(x \right) &= F_{4}\! \left(x \right) F_{98}\! \left(x \right)\\ F_{98}\! \left(x \right) &= F_{71}\! \left(x \right)+F_{99}\! \left(x \right)\\ F_{99}\! \left(x \right) &= F_{100}\! \left(x \right)\\ F_{100}\! \left(x \right) &= F_{101}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{101}\! \left(x \right) &= F_{102}\! \left(x \right)+F_{103}\! \left(x \right)\\ F_{102}\! \left(x \right) &= F_{2}\! \left(x \right)\\ F_{103}\! \left(x \right) &= F_{99}\! \left(x \right)\\ F_{104}\! \left(x \right) &= F_{105}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{105}\! \left(x \right) &= F_{106}\! \left(x \right)+F_{108}\! \left(x \right)\\ F_{106}\! \left(x \right) &= F_{107}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{107}\! \left(x \right) &= F_{50}\! \left(x \right)\\ F_{108}\! \left(x \right) &= F_{109}\! \left(x \right)+F_{99}\! \left(x \right)\\ F_{109}\! \left(x \right) &= F_{93}\! \left(x \right)\\ \end{align*}\)