Av(1324, 1432, 3241)
Generating Function
\(\displaystyle -\frac{5 x^{5}-19 x^{4}+25 x^{3}-19 x^{2}+7 x -1}{\left(x -1\right) \left(x^{2}-3 x +1\right) \left(3 x^{3}-5 x^{2}+4 x -1\right)}\)
Counting Sequence
1, 1, 2, 6, 21, 74, 249, 797, 2451, 7318, 21380, 61449, 174378, 489827, 1364499, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x -1\right) \left(x^{2}-3 x +1\right) \left(3 x^{3}-5 x^{2}+4 x -1\right) F \! \left(x \right)+5 x^{5}-19 x^{4}+25 x^{3}-19 x^{2}+7 x -1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 74\)
\(\displaystyle a \! \left(n +5\right) = 3 a \! \left(n \right)-14 a \! \left(n +1\right)+22 a \! \left(n +2\right)-18 a \! \left(n +3\right)+7 a \! \left(n +4\right)+2, \quad n \geq 6\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 74\)
\(\displaystyle a \! \left(n +5\right) = 3 a \! \left(n \right)-14 a \! \left(n +1\right)+22 a \! \left(n +2\right)-18 a \! \left(n +3\right)+7 a \! \left(n +4\right)+2, \quad n \geq 6\)
Explicit Closed Form
\(\displaystyle \frac{\left(6820 \left(\left(-\frac{129 \,\mathrm{I}}{62}+\frac{43 \sqrt{3}}{62}\right) \sqrt{31}+\mathrm{I} \sqrt{3}-1\right) 2^{\frac{2}{3}} \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}-750200+15035 \,2^{\frac{1}{3}} \left(\left(-\frac{399 \,\mathrm{I}}{3007}-\frac{133 \sqrt{3}}{3007}\right) \sqrt{31}+\mathrm{I} \sqrt{3}+1\right) \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{47 \left(\left(\mathrm{I}-\frac{9 \sqrt{31}}{47}\right) \sqrt{3}-\frac{27 \,\mathrm{I} \sqrt{31}}{47}+1\right) 2^{\frac{1}{3}} \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}}{8712}-\frac{\mathrm{I} \sqrt{3}\, \left(188+36 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{36}+\frac{\left(188+36 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{36}+\frac{5}{9}\right)^{-n}}{450120}+\frac{\left(-6820 \left(\left(-\frac{129 \,\mathrm{I}}{62}-\frac{43 \sqrt{3}}{62}\right) \sqrt{31}+\mathrm{I} \sqrt{3}+1\right) 2^{\frac{2}{3}} \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}-750200-15035 \,2^{\frac{1}{3}} \left(\left(-\frac{399 \,\mathrm{I}}{3007}+\frac{133 \sqrt{3}}{3007}\right) \sqrt{31}+\mathrm{I} \sqrt{3}-1\right) \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(-\frac{47 \left(\left(\mathrm{I}+\frac{9 \sqrt{31}}{47}\right) \sqrt{3}-\frac{27 \,\mathrm{I} \sqrt{31}}{47}-1\right) 2^{\frac{1}{3}} \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}}{8712}+\frac{\mathrm{I} \sqrt{3}\, \left(188+36 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{36}+\frac{\left(188+36 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{36}+\frac{5}{9}\right)^{-n}}{450120}+\frac{\left(\left(-9460 \sqrt{3}\, 2^{\frac{2}{3}} \sqrt{31}+13640 \,2^{\frac{2}{3}}\right) \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}-750200+\left(1330 \sqrt{31}\, \sqrt{3}\, 2^{\frac{1}{3}}-30070 \,2^{\frac{1}{3}}\right) \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}\right) \left(\frac{2^{\frac{1}{3}} \left(9 \sqrt{31}\, \sqrt{3}-47\right) \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{2}{3}}}{4356}-\frac{\left(188+36 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}}{18}+\frac{5}{9}\right)^{-n}}{450120}+\frac{\left(360096 \sqrt{5}+900240\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{450120}+2+\frac{\left(-360096 \sqrt{5}+900240\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{450120}\)
This specification was found using the strategy pack "Point Placements" and has 124 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_{17}\! \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_{5}\! \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_{11}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right) F_{13}\! \left(x \right)\\
F_{12}\! \left(x \right) &= x\\
F_{13}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{10}\! \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_{9}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{12}\! \left(x \right) F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{12}\! \left(x \right) F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{12}\! \left(x \right) F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{26}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{12}\! \left(x \right) F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{25}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{34}\! \left(x \right)+F_{75}\! \left(x \right)\\
F_{33}\! \left(x \right) &= 0\\
F_{34}\! \left(x \right) &= F_{12}\! \left(x \right) F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{38}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{12}\! \left(x \right) F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{12}\! \left(x \right) F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{40}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{12}\! \left(x \right) F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{50}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{12}\! \left(x \right) F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{12}\! \left(x \right) F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{46}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= 2 F_{33}\! \left(x \right)+F_{55}\! \left(x \right)+F_{65}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{12}\! \left(x \right) F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{53}\! \left(x \right)+F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)+F_{62}\! \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_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{63}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{12}\! \left(x \right) F_{64}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{57}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{12}\! \left(x \right) F_{66}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{61}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{68}\! \left(x \right)+F_{72}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{69}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{12}\! \left(x \right) F_{70}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{71}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{69}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{73}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{12}\! \left(x \right) F_{74}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{67}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{12}\! \left(x \right) F_{76}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{77}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{103}\! \left(x \right)+F_{78}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{33}\! \left(x \right)+F_{79}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{12}\! \left(x \right) F_{80}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{81}\! \left(x \right)+F_{92}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{82}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{83}\! \left(x \right)+F_{87}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{12}\! \left(x \right) F_{84}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{81}\! \left(x \right)+F_{85}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{86}\! \left(x \right)+F_{89}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{87}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{12}\! \left(x \right) F_{88}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{86}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{90}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{12}\! \left(x \right) F_{91}\! \left(x \right)\\
F_{91}\! \left(x \right) &= F_{85}\! \left(x \right)\\
F_{92}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{93}\! \left(x \right)\\
F_{93}\! \left(x \right) &= 2 F_{33}\! \left(x \right)+F_{94}\! \left(x \right)+F_{98}\! \left(x \right)\\
F_{94}\! \left(x \right) &= F_{12}\! \left(x \right) F_{95}\! \left(x \right)\\
F_{95}\! \left(x \right) &= F_{92}\! \left(x \right)+F_{96}\! \left(x \right)\\
F_{96}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{97}\! \left(x \right)\\
F_{97}\! \left(x \right) &= F_{98}\! \left(x \right)\\
F_{98}\! \left(x \right) &= F_{12}\! \left(x \right) F_{99}\! \left(x \right)\\
F_{99}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{97}\! \left(x \right)\\
F_{100}\! \left(x \right) &= F_{101}\! \left(x \right)\\
F_{101}\! \left(x \right) &= F_{102}\! \left(x \right) F_{12}\! \left(x \right)\\
F_{102}\! \left(x \right) &= F_{96}\! \left(x \right)\\
F_{103}\! \left(x \right) &= 2 F_{33}\! \left(x \right)+F_{104}\! \left(x \right)+F_{122}\! \left(x \right)\\
F_{104}\! \left(x \right) &= F_{105}\! \left(x \right) F_{12}\! \left(x \right)\\
F_{105}\! \left(x \right) &= F_{106}\! \left(x \right)+F_{112}\! \left(x \right)\\
F_{106}\! \left(x \right) &= F_{107}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{107}\! \left(x \right) &= 2 F_{33}\! \left(x \right)+F_{108}\! \left(x \right)+F_{110}\! \left(x \right)\\
F_{108}\! \left(x \right) &= F_{109}\! \left(x \right) F_{12}\! \left(x \right)\\
F_{109}\! \left(x \right) &= F_{106}\! \left(x \right)\\
F_{110}\! \left(x \right) &= F_{111}\! \left(x \right) F_{12}\! \left(x \right)\\
F_{111}\! \left(x \right) &= F_{46}\! \left(x \right)\\
F_{112}\! \left(x \right) &= F_{113}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{113}\! \left(x \right) &= 3 F_{33}\! \left(x \right)+F_{114}\! \left(x \right)+F_{116}\! \left(x \right)\\
F_{114}\! \left(x \right) &= F_{115}\! \left(x \right) F_{12}\! \left(x \right)\\
F_{115}\! \left(x \right) &= F_{112}\! \left(x \right)\\
F_{116}\! \left(x \right) &= F_{117}\! \left(x \right) F_{12}\! \left(x \right)\\
F_{117}\! \left(x \right) &= F_{118}\! \left(x \right)\\
F_{118}\! \left(x \right) &= F_{119}\! \left(x \right)+F_{71}\! \left(x \right)\\
F_{119}\! \left(x \right) &= F_{120}\! \left(x \right)\\
F_{120}\! \left(x \right) &= F_{12}\! \left(x \right) F_{121}\! \left(x \right)\\
F_{121}\! \left(x \right) &= F_{118}\! \left(x \right)\\
F_{122}\! \left(x \right) &= F_{12}\! \left(x \right) F_{123}\! \left(x \right)\\
F_{123}\! \left(x \right) &= F_{77}\! \left(x \right)\\
\end{align*}\)