Av(1342, 2143, 2341, 2413)
View Raw Data
Generating Function
\(\displaystyle \frac{x^{5}-4 x^{4}+9 x^{3}-12 x^{2}+6 x -1}{\left(2 x^{3}-4 x^{2}+4 x -1\right) \left(x^{2}-3 x +1\right)}\)
Counting Sequence
1, 1, 2, 6, 20, 65, 205, 634, 1935, 5847, 17526, 52183, 154503, 455278, 1336123, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(2 x^{3}-4 x^{2}+4 x -1\right) \left(x^{2}-3 x +1\right) F \! \left(x \right)-x^{5}+4 x^{4}-9 x^{3}+12 x^{2}-6 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 65\)
\(\displaystyle a \! \left(n +5\right) = 2 a \! \left(n \right)-10 a \! \left(n +1\right)+18 a \! \left(n +2\right)-17 a \! \left(n +3\right)+7 a \! \left(n +4\right), \quad n \geq 6\)
Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(-165 \,2^{\frac{2}{3}} \left(\left(\frac{\sqrt{11}}{33}+\mathrm{I}\right) \sqrt{3}+\frac{\mathrm{I} \sqrt{11}}{11}+1\right) \left(13+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}+10560-660 \left(\left(\frac{7 \sqrt{11}}{33}+\mathrm{I}\right) \sqrt{3}-\frac{7 \,\mathrm{I} \sqrt{11}}{11}-1\right) 2^{\frac{1}{3}} \left(13+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}\right) \left(\frac{13 \,2^{\frac{2}{3}} \left(\left(\mathrm{I}-\frac{3 \sqrt{11}}{13}\right) \sqrt{3}-\frac{9 \,\mathrm{I} \sqrt{11}}{13}+1\right) \left(13+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{384}-\frac{\mathrm{I} \sqrt{3}\, \left(26+6 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{\left(26+6 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{2}{3}\right)^{-n}}{21120}\\+\\\frac{\left(660 \,2^{\frac{1}{3}} \left(\left(-\frac{7 \sqrt{11}}{33}+\mathrm{I}\right) \sqrt{3}-\frac{7 \,\mathrm{I} \sqrt{11}}{11}+1\right) \left(13+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}+10560+165 \,2^{\frac{2}{3}} \left(\left(-\frac{\sqrt{11}}{33}+\mathrm{I}\right) \sqrt{3}+\frac{\mathrm{I} \sqrt{11}}{11}-1\right) \left(13+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(-\frac{13 \left(\left(\mathrm{I}+\frac{3 \sqrt{11}}{13}\right) \sqrt{3}-\frac{9 \,\mathrm{I} \sqrt{11}}{13}-1\right) 2^{\frac{2}{3}} \left(13+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{384}+\frac{\mathrm{I} \sqrt{3}\, \left(26+6 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{\left(26+6 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{12}+\frac{2}{3}\right)^{-n}}{21120}\\+\\\frac{\left(\left(280 \sqrt{3}\, 2^{\frac{1}{3}} \sqrt{11}-1320 \,2^{\frac{1}{3}}\right) \left(13+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}+10560+\left(10 \sqrt{3}\, 2^{\frac{2}{3}} \sqrt{11}+330 \,2^{\frac{2}{3}}\right) \left(13+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{2^{\frac{2}{3}} \left(3 \sqrt{11}\, \sqrt{3}-13\right) \left(13+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{192}-\frac{\left(26+6 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}+\frac{2}{3}\right)^{-n}}{21120}\\+\frac{\left(-2112 \sqrt{5}-10560\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{21120}+\frac{\left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n} \left(\sqrt{5}-5\right)}{10} & \text{otherwise} \end{array}\right.\)

This specification was found using the strategy pack "Point Placements" and has 139 rules.

Found on July 23, 2021.

Finding the specification took 7 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_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{13}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{65}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{0}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{34}\! \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_{13}\! \left(x \right) F_{14}\! \left(x \right)\\ F_{13}\! \left(x \right) &= x\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{18}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{16}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{13}\! \left(x \right) F_{15}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{19}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{21}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{20}\! \left(x \right) &= 0\\ F_{21}\! \left(x \right) &= F_{13}\! \left(x \right) F_{22}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{26}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{24}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{13}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{21}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{11}\! \left(x \right) F_{13}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{13}\! \left(x \right) F_{31}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{13}\! \left(x \right) F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{31}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{35}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{36}\! \left(x \right)+F_{38}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{13}\! \left(x \right) F_{37}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{16}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{13}\! \left(x \right) F_{39}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{47}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{42}\! \left(x \right)+F_{44}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{13}\! \left(x \right) F_{43}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{13}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{13}\! \left(x \right) F_{45}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{46}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{44}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{48}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{49}\! \left(x \right)+F_{51}\! \left(x \right)+F_{63}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{13}\! \left(x \right) F_{50}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{24}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{13}\! \left(x \right) F_{52}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)+F_{57}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)+F_{55}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{25}\! \left(x \right)+F_{42}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{13}\! \left(x \right) F_{46}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)+F_{60}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{49}\! \left(x \right)+F_{51}\! \left(x \right)+F_{59}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{13}\! \left(x \right) F_{35}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{61}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{13}\! \left(x \right) F_{62}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{63}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{13}\! \left(x \right) F_{64}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{62}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{133}\! \left(x \right) F_{66}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{107}\! \left(x \right)+F_{67}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{68}\! \left(x \right)+F_{71}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{69}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{13}\! \left(x \right) F_{70}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{68}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{72}\! \left(x \right)+F_{77}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{13}\! \left(x \right) F_{73}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{74}\! \left(x \right)\\ F_{74}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{72}\! \left(x \right)+F_{75}\! \left(x \right)\\ F_{75}\! \left(x \right) &= F_{13}\! \left(x \right) F_{76}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{68}\! \left(x \right)+F_{74}\! \left(x \right)\\ F_{77}\! \left(x \right) &= F_{13}\! \left(x \right) F_{78}\! \left(x \right)\\ F_{78}\! \left(x \right) &= F_{79}\! \left(x \right)+F_{88}\! \left(x \right)\\ F_{79}\! \left(x \right) &= F_{68}\! \left(x \right)+F_{80}\! \left(x \right)\\ F_{80}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{81}\! \left(x \right)+F_{85}\! \left(x \right)\\ F_{81}\! \left(x \right) &= F_{13}\! \left(x \right) F_{82}\! \left(x \right)\\ F_{82}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{83}\! \left(x \right)\\ F_{83}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{81}\! \left(x \right)+F_{84}\! \left(x \right)\\ F_{84}\! \left(x \right) &= F_{13}\! \left(x \right) F_{68}\! \left(x \right)\\ F_{85}\! \left(x \right) &= F_{13}\! \left(x \right) F_{86}\! \left(x \right)\\ F_{86}\! \left(x \right) &= F_{68}\! \left(x \right)+F_{87}\! \left(x \right)\\ F_{87}\! \left(x \right) &= F_{85}\! \left(x \right)\\ F_{88}\! \left(x \right) &= F_{71}\! \left(x \right)+F_{89}\! \left(x \right)\\ F_{89}\! \left(x \right) &= F_{105}\! \left(x \right)+F_{20}\! \left(x \right)+F_{90}\! \left(x \right)+F_{94}\! \left(x \right)\\ F_{90}\! \left(x \right) &= F_{13}\! \left(x \right) F_{91}\! \left(x \right)\\ F_{91}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{92}\! \left(x \right)\\ F_{92}\! \left(x \right) &= 2 F_{20}\! \left(x \right)+F_{90}\! \left(x \right)+F_{93}\! \left(x \right)\\ F_{93}\! \left(x \right) &= F_{13}\! \left(x \right) F_{74}\! \left(x \right)\\ F_{94}\! \left(x \right) &= F_{13}\! \left(x \right) F_{95}\! \left(x \right)\\ F_{95}\! \left(x \right) &= F_{96}\! \left(x \right)+F_{99}\! \left(x \right)\\ F_{96}\! \left(x \right) &= F_{83}\! \left(x \right)+F_{97}\! \left(x \right)\\ F_{97}\! \left(x \right) &= F_{98}\! \left(x \right)\\ F_{98}\! \left(x \right) &= F_{13}\! \left(x \right) F_{87}\! \left(x \right)\\ F_{99}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{102}\! \left(x \right)\\ F_{100}\! \left(x \right) &= F_{101}\! \left(x \right)+F_{20}\! \left(x \right)+F_{90}\! \left(x \right)+F_{94}\! \left(x \right)\\ F_{101}\! \left(x \right) &= F_{13}\! \left(x \right) F_{71}\! \left(x \right)\\ F_{102}\! \left(x \right) &= F_{103}\! \left(x \right)\\ F_{103}\! \left(x \right) &= F_{104}\! \left(x \right) F_{13}\! \left(x \right)\\ F_{104}\! \left(x \right) &= F_{105}\! \left(x \right)\\ F_{105}\! \left(x \right) &= F_{106}\! \left(x \right) F_{13}\! \left(x \right)\\ F_{106}\! \left(x \right) &= F_{104}\! \left(x \right)+F_{71}\! \left(x \right)\\ F_{107}\! \left(x \right) &= F_{108}\! \left(x \right)+F_{87}\! \left(x \right)\\ F_{108}\! \left(x \right) &= F_{109}\! \left(x \right)\\ F_{109}\! \left(x \right) &= F_{110}\! \left(x \right) F_{13}\! \left(x \right)\\ F_{110}\! \left(x \right) &= F_{111}\! \left(x \right)+F_{116}\! \left(x \right)\\ F_{111}\! \left(x \right) &= F_{112}\! \left(x \right)+F_{87}\! \left(x \right)\\ F_{112}\! \left(x \right) &= F_{113}\! \left(x \right)\\ F_{113}\! \left(x \right) &= F_{114}\! \left(x \right) F_{13}\! \left(x \right)\\ F_{114}\! \left(x \right) &= F_{115}\! \left(x \right)+F_{87}\! \left(x \right)\\ F_{115}\! \left(x \right) &= F_{113}\! \left(x \right)\\ F_{116}\! \left(x \right) &= F_{108}\! \left(x \right)+F_{117}\! \left(x \right)\\ F_{117}\! \left(x \right) &= 2 F_{20}\! \left(x \right)+F_{118}\! \left(x \right)+F_{119}\! \left(x \right)+F_{131}\! \left(x \right)\\ F_{118}\! \left(x \right) &= 0\\ F_{119}\! \left(x \right) &= F_{120}\! \left(x \right) F_{13}\! \left(x \right)\\ F_{120}\! \left(x \right) &= F_{121}\! \left(x \right)+F_{125}\! \left(x \right)\\ F_{121}\! \left(x \right) &= F_{122}\! \left(x \right)+F_{123}\! \left(x \right)\\ F_{122}\! \left(x \right) &= F_{98}\! \left(x \right)\\ F_{123}\! \left(x \right) &= F_{124}\! \left(x \right)\\ F_{124}\! \left(x \right) &= F_{115}\! \left(x \right) F_{13}\! \left(x \right)\\ F_{125}\! \left(x \right) &= F_{126}\! \left(x \right)+F_{128}\! \left(x \right)\\ F_{126}\! \left(x \right) &= 2 F_{20}\! \left(x \right)+F_{118}\! \left(x \right)+F_{119}\! \left(x \right)+F_{127}\! \left(x \right)\\ F_{127}\! \left(x \right) &= F_{108}\! \left(x \right) F_{13}\! \left(x \right)\\ F_{128}\! \left(x \right) &= F_{129}\! \left(x \right)\\ F_{129}\! \left(x \right) &= F_{13}\! \left(x \right) F_{130}\! \left(x \right)\\ F_{130}\! \left(x \right) &= F_{131}\! \left(x \right)\\ F_{131}\! \left(x \right) &= F_{13}\! \left(x \right) F_{132}\! \left(x \right)\\ F_{132}\! \left(x \right) &= F_{108}\! \left(x \right)+F_{130}\! \left(x \right)\\ F_{133}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{134}\! \left(x \right)\\ F_{134}\! \left(x \right) &= F_{135}\! \left(x \right)\\ F_{135}\! \left(x \right) &= F_{13}\! \left(x \right) F_{136}\! \left(x \right)\\ F_{136}\! \left(x \right) &= F_{133}\! \left(x \right)+F_{137}\! \left(x \right)\\ F_{137}\! \left(x \right) &= F_{138}\! \left(x \right)\\ F_{138}\! \left(x \right) &= F_{13}\! \left(x \right) F_{136}\! \left(x \right) F_{70}\! \left(x \right)\\ \end{align*}\)