Av(1243, 1342, 1432, 2341, 3142)
Generating Function
\(\displaystyle -\frac{3 x^{3}-5 x^{2}+4 x -1}{2 x^{4}-7 x^{3}+8 x^{2}-5 x +1}\)
Counting Sequence
1, 1, 2, 6, 19, 59, 181, 554, 1697, 5202, 15950, 48905, 149945, 459731, 1409530, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(2 x^{4}-7 x^{3}+8 x^{2}-5 x +1\right) F \! \left(x \right)+3 x^{3}-5 x^{2}+4 x -1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(n +4\right) = -2 a \! \left(n \right)+7 a \! \left(n +1\right)-8 a \! \left(n +2\right)+5 a \! \left(n +3\right), \quad n \geq 4\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(n +4\right) = -2 a \! \left(n \right)+7 a \! \left(n +1\right)-8 a \! \left(n +2\right)+5 a \! \left(n +3\right), \quad n \geq 4\)
Explicit Closed Form
\(\displaystyle \frac{154 \left(\underset{\alpha =\mathit{RootOf} \left(2 Z^{4}-7 Z^{3}+8 Z^{2}-5 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +2}\right)}{751}-\frac{207 \left(\underset{\alpha =\mathit{RootOf} \left(2 Z^{4}-7 Z^{3}+8 Z^{2}-5 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +1}\right)}{751}+\frac{199 \left(\underset{\alpha =\mathit{RootOf} \left(2 Z^{4}-7 Z^{3}+8 Z^{2}-5 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n}\right)}{751}+\frac{5 \left(\underset{\alpha =\mathit{RootOf} \left(2 Z^{4}-7 Z^{3}+8 Z^{2}-5 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n -1}\right)}{751}\)
This specification was found using the strategy pack "Point Placements" and has 102 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_{12}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{18}\! \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_{13}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{12}\! \left(x \right) &= x\\
F_{13}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{15}\! \left(x \right) &= 0\\
F_{16}\! \left(x \right) &= F_{10}\! \left(x \right) F_{12}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{12}\! \left(x \right) F_{13}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{12}\! \left(x \right) F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{26}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{12}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{12}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{28}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{12}\! \left(x \right) F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{12}\! \left(x \right) F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{12}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{36}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{12}\! \left(x \right) F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{39}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{31}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{35}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{12}\! \left(x \right) F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{12}\! \left(x \right) F_{19}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{12}\! \left(x \right) F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{15}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{12}\! \left(x \right) F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)+F_{55}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{51}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{12}\! \left(x \right) F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{12}\! \left(x \right) F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{13}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{57}\! \left(x \right)+F_{91}\! \left(x \right)+F_{98}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{12}\! \left(x \right) F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{59}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{50}\! \left(x \right)+F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{12}\! \left(x \right) F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{14}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{56}\! \left(x \right)+F_{64}\! \left(x \right)\\
F_{64}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{65}\! \left(x \right)+F_{69}\! \left(x \right)+F_{97}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{12}\! \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_{60}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{64}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{12}\! \left(x \right) F_{70}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{71}\! \left(x \right)\\
F_{71}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{72}\! \left(x \right)+F_{93}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{12}\! \left(x \right) F_{73}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{74}\! \left(x \right)+F_{95}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{12}\! \left(x \right) F_{75}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{76}\! \left(x \right)+F_{78}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{77}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{17}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{73}\! \left(x \right)+F_{79}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{80}\! \left(x \right)+F_{91}\! \left(x \right)+F_{93}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{12}\! \left(x \right) F_{81}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{82}\! \left(x \right)+F_{84}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{77}\! \left(x \right)+F_{83}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{61}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{79}\! \left(x \right)+F_{85}\! \left(x \right)\\
F_{85}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{69}\! \left(x \right)+F_{86}\! \left(x \right)+F_{90}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{12}\! \left(x \right) F_{87}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{88}\! \left(x \right)+F_{89}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{83}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{85}\! \left(x \right)\\
F_{90}\! \left(x \right) &= 0\\
F_{91}\! \left(x \right) &= F_{12}\! \left(x \right) F_{92}\! \left(x \right)\\
F_{92}\! \left(x \right) &= F_{71}\! \left(x \right)+F_{73}\! \left(x \right)\\
F_{93}\! \left(x \right) &= F_{12}\! \left(x \right) F_{94}\! \left(x \right)\\
F_{94}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{71}\! \left(x \right)\\
F_{95}\! \left(x \right) &= F_{12}\! \left(x \right) F_{96}\! \left(x \right)\\
F_{96}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{73}\! \left(x \right)\\
F_{97}\! \left(x \right) &= 0\\
F_{98}\! \left(x \right) &= F_{12}\! \left(x \right) F_{99}\! \left(x \right)\\
F_{99}\! \left(x \right) &= F_{94}\! \left(x \right)\\
F_{100}\! \left(x \right) &= F_{101}\! \left(x \right) F_{12}\! \left(x \right)\\
F_{101}\! \left(x \right) &= F_{94}\! \left(x \right)+F_{96}\! \left(x \right)\\
\end{align*}\)