Av(1342, 2341, 3142, 4123)
Generating Function
\(\displaystyle -\frac{5 x^{6}-24 x^{5}+43 x^{4}-44 x^{3}+26 x^{2}-8 x +1}{\left(x^{2}-3 x +1\right) \left(3 x^{3}-5 x^{2}+4 x -1\right) \left(-1+x \right)^{2}}\)
Counting Sequence
1, 1, 2, 6, 20, 65, 201, 597, 1726, 4904, 13770, 38329, 105951, 291189, 796366, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{2}-3 x +1\right) \left(3 x^{3}-5 x^{2}+4 x -1\right) \left(-1+x \right)^{2} F \! \left(x \right)+5 x^{6}-24 x^{5}+43 x^{4}-44 x^{3}+26 x^{2}-8 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 65\)
\(\displaystyle a \! \left(6\right) = 201\)
\(\displaystyle a \! \left(n +5\right) = 3 a \! \left(n \right)-14 a \! \left(n +1\right)+22 a \! \left(n +2\right)-18 a \! \left(n +3\right)+7 a \! \left(n +4\right)-n, \quad n \geq 7\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 65\)
\(\displaystyle a \! \left(6\right) = 201\)
\(\displaystyle a \! \left(n +5\right) = 3 a \! \left(n \right)-14 a \! \left(n +1\right)+22 a \! \left(n +2\right)-18 a \! \left(n +3\right)+7 a \! \left(n +4\right)-n, \quad n \geq 7\)
Explicit Closed Form
\(\displaystyle \frac{\left(-1705 \left(\left(\frac{51 \,\mathrm{I}}{31}-\frac{17 \sqrt{3}}{31}\right) \sqrt{31}+\mathrm{I} \sqrt{3}-1\right) 2^{\frac{2}{3}} \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}-150040+3565 \left(\left(-\frac{147 \,\mathrm{I}}{713}-\frac{49 \sqrt{3}}{713}\right) \sqrt{31}+\mathrm{I} \sqrt{3}+1\right) 2^{\frac{1}{3}} \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}}{225060}+\frac{\left(1705 \left(\left(\frac{51 \,\mathrm{I}}{31}+\frac{17 \sqrt{3}}{31}\right) \sqrt{31}+\mathrm{I} \sqrt{3}+1\right) 2^{\frac{2}{3}} \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}-150040-3565 \,2^{\frac{1}{3}} \left(\left(-\frac{147 \,\mathrm{I}}{713}+\frac{49 \sqrt{3}}{713}\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}}{225060}+\frac{\left(\left(-1870 \sqrt{3}\, 2^{\frac{2}{3}} \sqrt{31}-3410 \,2^{\frac{2}{3}}\right) \left(47+9 \sqrt{31}\, \sqrt{3}\right)^{\frac{1}{3}}-150040+\left(490 \sqrt{31}\, \sqrt{3}\, 2^{\frac{1}{3}}-7130 \,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}}{225060}+\frac{\left(90024 \sqrt{5}+225060\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{225060}+\frac{\left(-90024 \sqrt{5}+225060\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{225060}-n +1\)
This specification was found using the strategy pack "Point Placements" and has 95 rules.
Found on January 18, 2022.Finding the specification took 1 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_{35}\! \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_{23}\! \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_{22}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{18}\! \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_{26}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{26}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{30}\! \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_{33}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{27}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{30}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{14}\! \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_{41}\! \left(x \right)+F_{42}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{41}\! \left(x \right) &= 0\\
F_{42}\! \left(x \right) &= F_{4}\! \left(x \right) F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{40}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{39}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{47}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{4}\! \left(x \right) F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= 2 F_{41}\! \left(x \right)+F_{51}\! \left(x \right)+F_{54}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{4}\! \left(x \right) F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{50}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{4}\! \left(x \right) F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{56}\! \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_{26}\! \left(x \right)+F_{55}\! \left(x \right)\\
F_{59}\! \left(x \right) &= 2 F_{41}\! \left(x \right)+F_{54}\! \left(x \right)+F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{4}\! \left(x \right) F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{62}\! \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_{65}\! \left(x \right)+F_{66}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{67}\! \left(x \right)+F_{81}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{42}\! \left(x \right)+F_{68}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{4}\! \left(x \right) F_{69}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{70}\! \left(x \right)+F_{75}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{71}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{72}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{4}\! \left(x \right) F_{73}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{74}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{71}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{67}\! \left(x \right)+F_{76}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{77}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{4}\! \left(x \right) F_{78}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{79}\! \left(x \right)+F_{80}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{71}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{76}\! \left(x \right)\\
F_{81}\! \left(x \right) &= 2 F_{41}\! \left(x \right)+F_{60}\! \left(x \right)+F_{82}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{4}\! \left(x \right) F_{83}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{84}\! \left(x \right)+F_{89}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{56}\! \left(x \right)+F_{85}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{86}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{4}\! \left(x \right) F_{87}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{88}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{85}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{81}\! \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_{93}\! \left(x \right)+F_{94}\! \left(x \right)\\
F_{93}\! \left(x \right) &= F_{85}\! \left(x \right)\\
F_{94}\! \left(x \right) &= F_{90}\! \left(x \right)\\
\end{align*}\)