Av(1234, 1432, 2341, 3142, 3241)
View Raw Data
Generating Function
\(\displaystyle \frac{2 x^{8}+4 x^{7}+4 x^{6}-4 x^{5}-7 x^{4}-2 x^{3}+2 x -1}{\left(x -1\right) \left(x^{3}+x^{2}+x -1\right)^{2}}\)
Counting Sequence
1, 1, 2, 6, 19, 51, 120, 272, 595, 1267, 2650, 5462, 11127, 22455, 44960, ...
Implicit Equation for the Generating Function
\(\displaystyle -\left(x -1\right) \left(x^{3}+x^{2}+x -1\right)^{2} F \! \left(x \right)+2 x^{8}+4 x^{7}+4 x^{6}-4 x^{5}-7 x^{4}-2 x^{3}+2 x -1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 51\)
\(\displaystyle a \! \left(6\right) = 120\)
\(\displaystyle a \! \left(7\right) = 272\)
\(\displaystyle a \! \left(8\right) = 595\)
\(\displaystyle a \! \left(n +2\right) = -\frac{a \! \left(n \right)}{3}-\frac{2 a \! \left(n +1\right)}{3}+\frac{a \! \left(n +4\right)}{3}+\frac{2 a \! \left(n +5\right)}{3}-\frac{a \! \left(n +6\right)}{3}+\frac{2}{3}, \quad n \geq 9\)
Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0 \\ 1 & n =1 \\ \frac{\left(\left(-462 \left(\mathrm{I}+\frac{\sqrt{3}}{3}\right) \left(n -\frac{8}{11}\right) \sqrt{11}+770 \left(n -\frac{6}{35}\right) \left(1+\mathrm{I} \sqrt{3}\right)\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}+\left(-231 \left(n -\frac{377}{77}\right) \left(\mathrm{I}-\frac{\sqrt{3}}{3}\right) \sqrt{11}+539 \left(\mathrm{I} \sqrt{3}-1\right) \left(n -\frac{201}{49}\right)\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}+1408 n -2904\right) \left(\frac{\left(\left(17 \,\mathrm{I}+3 \sqrt{11}\right) \sqrt{3}-9 \,\mathrm{I} \sqrt{11}-17\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{24}-\frac{\mathrm{I} \sqrt{3}\, \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{1}{3}\right)^{-n}}{5808}\\+\\\frac{\left(\left(462 \left(n -\frac{8}{11}\right) \left(\mathrm{I}-\frac{\sqrt{3}}{3}\right) \sqrt{11}-770 \left(n -\frac{6}{35}\right) \left(\mathrm{I} \sqrt{3}-1\right)\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}+\left(231 \left(\mathrm{I}+\frac{\sqrt{3}}{3}\right) \left(n -\frac{377}{77}\right) \sqrt{11}-539 \left(1+\mathrm{I} \sqrt{3}\right) \left(n -\frac{201}{49}\right)\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}+1408 n -2904\right) \left(\frac{\left(\left(-17 \,\mathrm{I}+3 \sqrt{11}\right) \sqrt{3}+9 \,\mathrm{I} \sqrt{11}-17\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{24}+\frac{\mathrm{I} \sqrt{3}\, \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{6}-\frac{1}{3}\right)^{-n}}{5808}\\+\frac{1}{2}+\\\frac{\left(\left(\left(308 n -224\right) \sqrt{3}\, \sqrt{11}-1540 n +264\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}+\left(\left(-154 n +754\right) \sqrt{3}\, \sqrt{11}+1078 n -4422\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}+1408 n -2904\right) \left(\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}}{3}-\frac{1}{3}+\frac{17 \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}}{12}-\frac{\left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}} \sqrt{11}\, \sqrt{3}}{4}\right)^{-n}}{5808} & \text{otherwise} \end{array}\right.\)

This specification was found using the strategy pack "Point Placements" and has 110 rules.

Found on January 18, 2022.

Finding the specification took 2 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_{26}\! \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_{14}\! \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_{4}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{17}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{13}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{21}\! \left(x \right)\\ F_{18}\! \left(x \right) &= 0\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{11}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{4}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{28}\! \left(x \right)+F_{65}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x \right) F_{4}\! \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_{7}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{33}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{35}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{11}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{19}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{38}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{28}\! \left(x \right)\\ F_{38}\! \left(x \right) &= 2 F_{18}\! \left(x \right)+F_{39}\! \left(x \right)+F_{43}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{4}\! \left(x \right) F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{42}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{31}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{39}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{4}\! \left(x \right) F_{44}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{57}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)\\ F_{46}\! \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_{49}\! \left(x \right)+F_{51}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{50}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{32}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{52}\! \left(x \right)\\ F_{52}\! \left(x \right) &= 2 F_{18}\! \left(x \right)+F_{43}\! \left(x \right)+F_{53}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{4}\! \left(x \right) F_{54}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)+F_{56}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{50}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{53}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)\\ F_{58}\! \left(x \right) &= 2 F_{18}\! \left(x \right)+F_{59}\! \left(x \right)+F_{63}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{4}\! \left(x \right) F_{60}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{61}\! \left(x \right)+F_{62}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{19}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{59}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{4}\! \left(x \right) F_{64}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{46}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{4}\! \left(x \right) F_{66}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{67}\! \left(x \right)+F_{68}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{46}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{109}\! \left(x \right)+F_{69}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{107}\! \left(x \right)+F_{18}\! \left(x \right)+F_{70}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{4}\! \left(x \right) F_{71}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{72}\! \left(x \right)+F_{78}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{73}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{74}\! \left(x \right)\\ F_{74}\! \left(x \right) &= F_{4}\! \left(x \right) F_{75}\! \left(x \right)\\ F_{75}\! \left(x \right) &= F_{76}\! \left(x \right)+F_{77}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{4}\! \left(x \right)\\ F_{77}\! \left(x \right) &= F_{23}\! \left(x \right)\\ F_{78}\! \left(x \right) &= F_{79}\! \left(x \right)+F_{80}\! \left(x \right)\\ F_{79}\! \left(x \right) &= F_{70}\! \left(x \right)\\ F_{80}\! \left(x \right) &= 2 F_{18}\! \left(x \right)+F_{81}\! \left(x \right)+F_{85}\! \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_{84}\! \left(x \right)\\ F_{83}\! \left(x \right) &= F_{73}\! \left(x \right)\\ F_{84}\! \left(x \right) &= F_{81}\! \left(x \right)\\ F_{85}\! \left(x \right) &= F_{4}\! \left(x \right) F_{86}\! \left(x \right)\\ F_{86}\! \left(x \right) &= F_{87}\! \left(x \right)+F_{99}\! \left(x \right)\\ F_{87}\! \left(x \right) &= F_{88}\! \left(x \right)\\ F_{88}\! \left(x \right) &= F_{89}\! \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_{93}\! \left(x \right)\\ F_{91}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{92}\! \left(x \right)\\ F_{92}\! \left(x \right) &= F_{74}\! \left(x \right)\\ F_{93}\! \left(x \right) &= F_{88}\! \left(x \right)+F_{94}\! \left(x \right)\\ F_{94}\! \left(x \right) &= 2 F_{18}\! \left(x \right)+F_{85}\! \left(x \right)+F_{95}\! \left(x \right)\\ F_{95}\! \left(x \right) &= F_{4}\! \left(x \right) F_{96}\! \left(x \right)\\ F_{96}\! \left(x \right) &= F_{97}\! \left(x \right)+F_{98}\! \left(x \right)\\ F_{97}\! \left(x \right) &= F_{92}\! \left(x \right)\\ F_{98}\! \left(x \right) &= F_{95}\! \left(x \right)\\ F_{99}\! \left(x \right) &= F_{100}\! \left(x \right)\\ F_{100}\! \left(x \right) &= 2 F_{18}\! \left(x \right)+F_{101}\! \left(x \right)+F_{105}\! \left(x \right)\\ F_{101}\! \left(x \right) &= F_{102}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{102}\! \left(x \right) &= F_{103}\! \left(x \right)+F_{104}\! \left(x \right)\\ F_{103}\! \left(x \right) &= F_{23}\! \left(x \right)\\ F_{104}\! \left(x \right) &= F_{101}\! \left(x \right)\\ F_{105}\! \left(x \right) &= F_{106}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{106}\! \left(x \right) &= F_{88}\! \left(x \right)\\ F_{107}\! \left(x \right) &= F_{108}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{108}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{88}\! \left(x \right)\\ F_{109}\! \left(x \right) &= 2 F_{18}\! \left(x \right)+F_{59}\! \left(x \right)+F_{63}\! \left(x \right)\\ \end{align*}\)