Av(1243, 1342, 1423, 3241)
Generating Function
\(\displaystyle -\frac{3 x^{5}-7 x^{4}+8 x^{3}-9 x^{2}+5 x -1}{\left(2 x^{3}-x^{2}+3 x -1\right) \left(x -1\right)^{3}}\)
Counting Sequence
1, 1, 2, 6, 20, 64, 195, 578, 1691, 4917, 14257, 41287, 119500, 345801, 1000564, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(2 x^{3}-x^{2}+3 x -1\right) \left(x -1\right)^{3} F \! \left(x \right)+3 x^{5}-7 x^{4}+8 x^{3}-9 x^{2}+5 x -1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 64\)
\(\displaystyle a \! \left(n +3\right) = \frac{n^{2}}{2}+2 a \! \left(n \right)-a \! \left(n +1\right)+3 a \! \left(n +2\right)+\frac{5 n}{2}-1, \quad n \geq 6\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 64\)
\(\displaystyle a \! \left(n +3\right) = \frac{n^{2}}{2}+2 a \! \left(n \right)-a \! \left(n +1\right)+3 a \! \left(n +2\right)+\frac{5 n}{2}-1, \quad n \geq 6\)
Explicit Closed Form
\(\displaystyle \frac{\left(\left(\left(-43809 \,\mathrm{I}-14603 \sqrt{3}\right) \sqrt{211}+104023 \,\mathrm{I} \sqrt{3}+104023\right) \left(28+3 \sqrt{211}\, \sqrt{3}\right)^{\frac{1}{3}}+975664+\left(\left(-1005 \,\mathrm{I}+335 \sqrt{3}\right) \sqrt{211}+85877 \,\mathrm{I} \sqrt{3}-85877\right) \left(28+3 \sqrt{211}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{\left(\left(28 \,\mathrm{I}+3 \sqrt{211}\right) \sqrt{3}-9 \,\mathrm{I} \sqrt{211}-28\right) \left(28+3 \sqrt{211}\, \sqrt{3}\right)^{\frac{2}{3}}}{3468}-\frac{\mathrm{I} \sqrt{3}\, \left(28+3 \sqrt{211}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}-\frac{\left(28+3 \sqrt{211}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{1}{6}\right)^{-n}}{19757196}+\frac{\left(\left(\left(43809 \,\mathrm{I}-14603 \sqrt{3}\right) \sqrt{211}-104023 \,\mathrm{I} \sqrt{3}+104023\right) \left(28+3 \sqrt{211}\, \sqrt{3}\right)^{\frac{1}{3}}+975664+\left(\left(1005 \,\mathrm{I}+335 \sqrt{3}\right) \sqrt{211}-85877 \,\mathrm{I} \sqrt{3}-85877\right) \left(28+3 \sqrt{211}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{\left(\left(-28 \,\mathrm{I}+3 \sqrt{211}\right) \sqrt{3}+9 \,\mathrm{I} \sqrt{211}-28\right) \left(28+3 \sqrt{211}\, \sqrt{3}\right)^{\frac{2}{3}}}{3468}+\frac{\mathrm{I} \sqrt{3}\, \left(28+3 \sqrt{211}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}-\frac{\left(28+3 \sqrt{211}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{1}{6}\right)^{-n}}{19757196}+\frac{\left(\left(29206 \sqrt{211}\, \sqrt{3}-208046\right) \left(28+3 \sqrt{211}\, \sqrt{3}\right)^{\frac{1}{3}}-670 \left(28+3 \sqrt{211}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{211}\, \sqrt{3}+171754 \left(28+3 \sqrt{211}\, \sqrt{3}\right)^{\frac{2}{3}}+975664\right) \left(\frac{\left(28+3 \sqrt{211}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}+\frac{1}{6}+\frac{14 \left(28+3 \sqrt{211}\, \sqrt{3}\right)^{\frac{2}{3}}}{867}-\frac{\left(28+3 \sqrt{211}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{211}\, \sqrt{3}}{578}\right)^{-n}}{19757196}-\frac{n^{2}}{6}-\frac{11 n}{18}+\frac{23}{27}\)
This specification was found using the strategy pack "Point Placements" and has 146 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_{12}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{20}\! \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_{13}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{12}\! \left(x \right) &= x\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{12}\! \left(x \right) F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{12}\! \left(x \right) F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{12}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{12}\! \left(x \right) F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{14}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \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_{36}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{12}\! \left(x \right) F_{33}\! \left(x \right)\\
F_{36}\! \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_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{42}\! \left(x \right)+F_{93}\! \left(x \right)\\
F_{41}\! \left(x \right) &= 0\\
F_{42}\! \left(x \right) &= F_{12}\! \left(x \right) F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{61}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{46}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{12}\! \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_{10}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{12}\! \left(x \right) F_{48}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{17}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{12}\! \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_{55}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{12}\! \left(x \right) F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{55}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{12}\! \left(x \right) F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= 2 F_{41}\! \left(x \right)+F_{63}\! \left(x \right)+F_{84}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{12}\! \left(x \right) F_{64}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{65}\! \left(x \right)+F_{83}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{66}\! \left(x \right)+F_{81}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{67}\! \left(x \right)+F_{79}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{12}\! \left(x \right) F_{68}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{69}\! \left(x \right)+F_{72}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{70}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{46}\! \left(x \right)+F_{71}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{12}\! \left(x \right) F_{13}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{66}\! \left(x \right)+F_{73}\! \left(x \right)\\
F_{73}\! \left(x \right) &= 2 F_{41}\! \left(x \right)+F_{63}\! \left(x \right)+F_{74}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{12}\! \left(x \right) F_{75}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{76}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{77}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{12}\! \left(x \right) F_{78}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{66}\! \left(x \right)+F_{76}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{12}\! \left(x \right) F_{80}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{66}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{82}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{12}\! \left(x \right) F_{65}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{76}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{12}\! \left(x \right) F_{85}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{75}\! \left(x \right)+F_{86}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{87}\! \left(x \right)+F_{90}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{88}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{12}\! \left(x \right) F_{89}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{87}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{91}\! \left(x \right)\\
F_{91}\! \left(x \right) &= F_{12}\! \left(x \right) F_{92}\! \left(x \right)\\
F_{92}\! \left(x \right) &= F_{76}\! \left(x \right)+F_{90}\! \left(x \right)\\
F_{93}\! \left(x \right) &= F_{12}\! \left(x \right) F_{94}\! \left(x \right)\\
F_{94}\! \left(x \right) &= F_{80}\! \left(x \right)+F_{95}\! \left(x \right)\\
F_{95}\! \left(x \right) &= F_{120}\! \left(x \right)+F_{96}\! \left(x \right)\\
F_{96}\! \left(x \right) &= F_{118}\! \left(x \right)+F_{41}\! \left(x \right)+F_{97}\! \left(x \right)\\
F_{97}\! \left(x \right) &= F_{12}\! \left(x \right) F_{98}\! \left(x \right)\\
F_{98}\! \left(x \right) &= F_{106}\! \left(x \right)+F_{99}\! \left(x \right)\\
F_{99}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{100}\! \left(x \right) &= F_{101}\! \left(x \right)+F_{41}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{101}\! \left(x \right) &= F_{102}\! \left(x \right) F_{12}\! \left(x \right)\\
F_{102}\! \left(x \right) &= F_{103}\! \left(x \right)\\
F_{103}\! \left(x \right) &= F_{104}\! \left(x \right)+F_{12}\! \left(x \right)\\
F_{104}\! \left(x \right) &= F_{105}\! \left(x \right)\\
F_{105}\! \left(x \right) &= F_{103}\! \left(x \right) F_{12}\! \left(x \right)\\
F_{106}\! \left(x \right) &= F_{107}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{107}\! \left(x \right) &= 2 F_{41}\! \left(x \right)+F_{108}\! \left(x \right)+F_{115}\! \left(x \right)\\
F_{108}\! \left(x \right) &= F_{109}\! \left(x \right) F_{12}\! \left(x \right)\\
F_{109}\! \left(x \right) &= F_{110}\! \left(x \right)\\
F_{110}\! \left(x \right) &= F_{111}\! \left(x \right)+F_{113}\! \left(x \right)\\
F_{111}\! \left(x \right) &= F_{112}\! \left(x \right)\\
F_{112}\! \left(x \right) &= F_{12}\! \left(x \right) F_{21}\! \left(x \right)\\
F_{113}\! \left(x \right) &= F_{114}\! \left(x \right)\\
F_{114}\! \left(x \right) &= F_{110}\! \left(x \right) F_{12}\! \left(x \right)\\
F_{115}\! \left(x \right) &= F_{116}\! \left(x \right) F_{12}\! \left(x \right)\\
F_{116}\! \left(x \right) &= F_{117}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{117}\! \left(x \right) &= F_{115}\! \left(x \right)\\
F_{118}\! \left(x \right) &= F_{119}\! \left(x \right) F_{12}\! \left(x \right)\\
F_{119}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{96}\! \left(x \right)\\
F_{120}\! \left(x \right) &= 2 F_{41}\! \left(x \right)+F_{121}\! \left(x \right)+F_{144}\! \left(x \right)\\
F_{121}\! \left(x \right) &= F_{12}\! \left(x \right) F_{122}\! \left(x \right)\\
F_{122}\! \left(x \right) &= F_{123}\! \left(x \right)+F_{132}\! \left(x \right)\\
F_{123}\! \left(x \right) &= F_{124}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{124}\! \left(x \right) &= 2 F_{41}\! \left(x \right)+F_{125}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{125}\! \left(x \right) &= F_{12}\! \left(x \right) F_{126}\! \left(x \right)\\
F_{126}\! \left(x \right) &= F_{127}\! \left(x \right)\\
F_{127}\! \left(x \right) &= F_{128}\! \left(x \right)+F_{130}\! \left(x \right)\\
F_{128}\! \left(x \right) &= F_{129}\! \left(x \right)\\
F_{129}\! \left(x \right) &= F_{10}\! \left(x \right) F_{12}\! \left(x \right)\\
F_{130}\! \left(x \right) &= F_{131}\! \left(x \right)\\
F_{131}\! \left(x \right) &= F_{12}\! \left(x \right) F_{127}\! \left(x \right)\\
F_{132}\! \left(x \right) &= F_{133}\! \left(x \right)+F_{76}\! \left(x \right)\\
F_{133}\! \left(x \right) &= 3 F_{41}\! \left(x \right)+F_{134}\! \left(x \right)+F_{141}\! \left(x \right)\\
F_{134}\! \left(x \right) &= F_{12}\! \left(x \right) F_{135}\! \left(x \right)\\
F_{135}\! \left(x \right) &= F_{136}\! \left(x \right)\\
F_{136}\! \left(x \right) &= F_{137}\! \left(x \right)+F_{139}\! \left(x \right)\\
F_{137}\! \left(x \right) &= F_{138}\! \left(x \right)\\
F_{138}\! \left(x \right) &= F_{12}\! \left(x \right) F_{66}\! \left(x \right)\\
F_{139}\! \left(x \right) &= F_{140}\! \left(x \right)\\
F_{140}\! \left(x \right) &= F_{12}\! \left(x \right) F_{136}\! \left(x \right)\\
F_{141}\! \left(x \right) &= F_{12}\! \left(x \right) F_{142}\! \left(x \right)\\
F_{142}\! \left(x \right) &= F_{143}\! \left(x \right)+F_{76}\! \left(x \right)\\
F_{143}\! \left(x \right) &= F_{141}\! \left(x \right)\\
F_{144}\! \left(x \right) &= F_{12}\! \left(x \right) F_{145}\! \left(x \right)\\
F_{145}\! \left(x \right) &= F_{120}\! \left(x \right)+F_{66}\! \left(x \right)\\
\end{align*}\)