Av(1342, 3214, 3412, 4123)
Generating Function
\(\displaystyle -\frac{2 x^{9}-4 x^{8}+x^{7}+x^{6}+6 x^{5}-6 x^{4}+8 x^{3}-10 x^{2}+5 x -1}{\left(x^{3}+x^{2}+x -1\right) \left(x -1\right)^{5}}\)
Counting Sequence
1, 1, 2, 6, 20, 58, 143, 315, 647, 1269, 2416, 4516, 8350, 15348, 28133, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{3}+x^{2}+x -1\right) \left(x -1\right)^{5} F \! \left(x \right)+2 x^{9}-4 x^{8}+x^{7}+x^{6}+6 x^{5}-6 x^{4}+8 x^{3}-10 x^{2}+5 x -1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 58\)
\(\displaystyle a \! \left(6\right) = 143\)
\(\displaystyle a \! \left(7\right) = 315\)
\(\displaystyle a \! \left(8\right) = 647\)
\(\displaystyle a \! \left(9\right) = 1269\)
\(\displaystyle a \! \left(n +3\right) = -\frac{n^{4}}{12}+\frac{n^{3}}{2}+\frac{37 n^{2}}{12}+a \! \left(n \right)+a \! \left(n +1\right)+a \! \left(n +2\right)+\frac{19 n}{2}-4, \quad n \geq 10\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 58\)
\(\displaystyle a \! \left(6\right) = 143\)
\(\displaystyle a \! \left(7\right) = 315\)
\(\displaystyle a \! \left(8\right) = 647\)
\(\displaystyle a \! \left(9\right) = 1269\)
\(\displaystyle a \! \left(n +3\right) = -\frac{n^{4}}{12}+\frac{n^{3}}{2}+\frac{37 n^{2}}{12}+a \! \left(n \right)+a \! \left(n +1\right)+a \! \left(n +2\right)+\frac{19 n}{2}-4, \quad n \geq 10\)
Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0 \\ 1 & n =1 \\ \frac{\left(\left(\left(-36 \sqrt{11}+132 \,\mathrm{I}\right) \sqrt{3}-108 \,\mathrm{I} \sqrt{11}+132\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}+528+\left(\left(54 \sqrt{11}+330 \,\mathrm{I}\right) \sqrt{3}-162 \,\mathrm{I} \sqrt{11}-330\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}\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}}{264}\\+\\\frac{\left(\left(\left(54 \sqrt{11}-330 \,\mathrm{I}\right) \sqrt{3}+162 \,\mathrm{I} \sqrt{11}-330\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}+528+\left(\left(-36 \sqrt{11}-132 \,\mathrm{I}\right) \sqrt{3}+108 \,\mathrm{I} \sqrt{11}+132\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}\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}}{264}\\+\\\frac{\left(\left(-108 \sqrt{11}\, \sqrt{3}+660\right) \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{2}{3}}+72 \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}} \sqrt{11}\, \sqrt{3}-264 \left(17+3 \sqrt{11}\, \sqrt{3}\right)^{\frac{1}{3}}+528\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}}{264}\\+\frac{n^{4}}{24}-\frac{n^{3}}{4}-\frac{25 n^{2}}{24}-\frac{19 n}{4}-1 & \text{otherwise} \end{array}\right.\)
This specification was found using the strategy pack "Point Placements" and has 131 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_{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_{6}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{22}\! \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_{14}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{14}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{7}\! \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_{7}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{30}\! \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_{14}\! \left(x \right)+F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{36}\! \left(x \right)+F_{48}\! \left(x \right)\\
F_{35}\! \left(x \right) &= 0\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{4}\! \left(x \right) F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{39}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{40}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{4}\! \left(x \right) F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{45}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{4}\! \left(x \right) F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{48}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{38}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{110}\! \left(x \right)+F_{35}\! \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_{71}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{35}\! \left(x \right)+F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{4}\! \left(x \right) F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{59}\! \left(x \right)+F_{60}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{26}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{61}\! \left(x \right)+F_{68}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{62}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{21}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{4}\! \left(x \right) F_{64}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{65}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{66}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{62}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{61}\! \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_{26}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{72}\! \left(x \right)+F_{93}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{73}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{4}\! \left(x \right) F_{74}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{75}\! \left(x \right)+F_{77}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{76}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{73}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{78}\! \left(x \right)+F_{86}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{79}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{4}\! \left(x \right) F_{80}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{81}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{78}\! \left(x \right)+F_{82}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{83}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{4}\! \left(x \right) F_{84}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{85}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{39}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{87}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{4}\! \left(x \right) F_{88}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{89}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{76}\! \left(x \right)+F_{90}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{91}\! \left(x \right)\\
F_{91}\! \left(x \right) &= F_{4}\! \left(x \right) F_{92}\! \left(x \right)\\
F_{92}\! \left(x \right) &= F_{76}\! \left(x \right)\\
F_{93}\! \left(x \right) &= 2 F_{35}\! \left(x \right)+F_{94}\! \left(x \right)+F_{96}\! \left(x \right)\\
F_{94}\! \left(x \right) &= F_{4}\! \left(x \right) F_{95}\! \left(x \right)\\
F_{95}\! \left(x \right) &= F_{76}\! \left(x \right)\\
F_{96}\! \left(x \right) &= F_{4}\! \left(x \right) F_{97}\! \left(x \right)\\
F_{97}\! \left(x \right) &= F_{98}\! \left(x \right)+F_{99}\! \left(x \right)\\
F_{98}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{90}\! \left(x \right)\\
F_{99}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{107}\! \left(x \right)\\
F_{100}\! \left(x \right) &= 2 F_{35}\! \left(x \right)+F_{101}\! \left(x \right)+F_{102}\! \left(x \right)\\
F_{101}\! \left(x \right) &= F_{4}\! \left(x \right) F_{85}\! \left(x \right)\\
F_{102}\! \left(x \right) &= F_{103}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{103}\! \left(x \right) &= F_{104}\! \left(x \right)+F_{106}\! \left(x \right)\\
F_{104}\! \left(x \right) &= F_{105}\! \left(x \right)+F_{39}\! \left(x \right)\\
F_{105}\! \left(x \right) &= F_{101}\! \left(x \right)\\
F_{106}\! \left(x \right) &= F_{100}\! \left(x \right)\\
F_{107}\! \left(x \right) &= F_{108}\! \left(x \right)\\
F_{108}\! \left(x \right) &= F_{109}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{109}\! \left(x \right) &= F_{90}\! \left(x \right)\\
F_{110}\! \left(x \right) &= F_{111}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{111}\! \left(x \right) &= F_{112}\! \left(x \right)+F_{116}\! \left(x \right)\\
F_{112}\! \left(x \right) &= F_{113}\! \left(x \right)+F_{15}\! \left(x \right)\\
F_{113}\! \left(x \right) &= F_{110}\! \left(x \right)+F_{114}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{114}\! \left(x \right) &= F_{115}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{115}\! \left(x \right) &= F_{7}\! \left(x \right)+F_{76}\! \left(x \right)\\
F_{116}\! \left(x \right) &= F_{117}\! \left(x \right)+F_{124}\! \left(x \right)\\
F_{117}\! \left(x \right) &= F_{118}\! \left(x \right)+F_{35}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{118}\! \left(x \right) &= F_{119}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{119}\! \left(x \right) &= F_{120}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{120}\! \left(x \right) &= F_{117}\! \left(x \right)+F_{121}\! \left(x \right)\\
F_{121}\! \left(x \right) &= F_{122}\! \left(x \right)\\
F_{122}\! \left(x \right) &= F_{123}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{123}\! \left(x \right) &= F_{39}\! \left(x \right)\\
F_{124}\! \left(x \right) &= 2 F_{35}\! \left(x \right)+F_{125}\! \left(x \right)+F_{127}\! \left(x \right)\\
F_{125}\! \left(x \right) &= F_{126}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{126}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{90}\! \left(x \right)\\
F_{127}\! \left(x \right) &= F_{128}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{128}\! \left(x \right) &= F_{129}\! \left(x \right)\\
F_{129}\! \left(x \right) &= F_{130}\! \left(x \right)+F_{76}\! \left(x \right)\\
F_{130}\! \left(x \right) &= F_{94}\! \left(x \right)\\
\end{align*}\)