Av(12453, 12543, 13452, 13542, 14352, 21453, 21543, 23451, 23541, 24351, 31452, 31542, 32451, 32541, 34251, 41352, 42351, 43251)
Generating Function
\(\displaystyle \frac{5 x^{6}-14 x^{5}+32 x^{4}-39 x^{3}+28 x^{2}-9 x +1}{\left(6 x^{3}-9 x^{2}+6 x -1\right) \left(x^{3}-3 x^{2}+4 x -1\right)}\)
Counting Sequence
1, 1, 2, 6, 24, 102, 434, 1838, 7760, 32704, 137670, 579056, 2434086, 10227078, 42955358, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(6 x^{3}-9 x^{2}+6 x -1\right) \left(x^{3}-3 x^{2}+4 x -1\right) F \! \left(x \right)-5 x^{6}+14 x^{5}-32 x^{4}+39 x^{3}-28 x^{2}+9 x -1 = 0\)
Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 102\)
\(\displaystyle a(6) = 434\)
\(\displaystyle a{\left(n + 6 \right)} = - 6 a{\left(n \right)} + 27 a{\left(n + 1 \right)} - 57 a{\left(n + 2 \right)} + 61 a{\left(n + 3 \right)} - 36 a{\left(n + 4 \right)} + 10 a{\left(n + 5 \right)}, \quad n \geq 7\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 102\)
\(\displaystyle a(6) = 434\)
\(\displaystyle a{\left(n + 6 \right)} = - 6 a{\left(n \right)} + 27 a{\left(n + 1 \right)} - 57 a{\left(n + 2 \right)} + 61 a{\left(n + 3 \right)} - 36 a{\left(n + 4 \right)} + 10 a{\left(n + 5 \right)}, \quad n \geq 7\)
Explicit Closed Form
\(\displaystyle -\frac{35688 \left(\underset{\alpha =\mathit{RootOf} \left(6 Z^{6}-27 Z^{5}+57 Z^{4}-61 Z^{3}+36 Z^{2}-10 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +4}\right)}{87079}+\frac{153268 \left(\underset{\alpha =\mathit{RootOf} \left(6 Z^{6}-27 Z^{5}+57 Z^{4}-61 Z^{3}+36 Z^{2}-10 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +3}\right)}{87079}-\frac{845746 \left(\underset{\alpha =\mathit{RootOf} \left(6 Z^{6}-27 Z^{5}+57 Z^{4}-61 Z^{3}+36 Z^{2}-10 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +2}\right)}{261237}+\frac{271978 \left(\underset{\alpha =\mathit{RootOf} \left(6 Z^{6}-27 Z^{5}+57 Z^{4}-61 Z^{3}+36 Z^{2}-10 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +1}\right)}{87079}-\frac{372598 \left(\underset{\alpha =\mathit{RootOf} \left(6 Z^{6}-27 Z^{5}+57 Z^{4}-61 Z^{3}+36 Z^{2}-10 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n}\right)}{261237}+\frac{57622 \left(\underset{\alpha =\mathit{RootOf} \left(6 Z^{6}-27 Z^{5}+57 Z^{4}-61 Z^{3}+36 Z^{2}-10 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n -1}\right)}{261237}+\left(\left\{\begin{array}{cc}\frac{5}{6} & n =0 \\ 0 & \text{otherwise} \end{array}\right.\right)\)
This specification was found using the strategy pack "Regular Insertion Encoding Bottom" and has 94 rules.
Finding the specification took 269 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_{23}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)+F_{8}\! \left(x \right)+F_{86}\! \left(x \right)\\
F_{7}\! \left(x \right) &= 0\\
F_{8}\! \left(x \right) &= F_{23}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{81}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{44}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{23}\! \left(x \right) &= x\\
F_{24}\! \left(x \right) &= F_{20}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= 2 F_{7}\! \left(x \right)+F_{27}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{23}\! \left(x \right) F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{23}\! \left(x \right) F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{24}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{30}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{23}\! \left(x \right) F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{26}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= 2 F_{7}\! \left(x \right)+F_{35}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{23}\! \left(x \right) F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{31}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{23}\! \left(x \right) F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{71}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)+F_{50}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{23}\! \left(x \right) F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{23}\! \left(x \right) F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)+F_{55}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{50}\! \left(x \right)+F_{54}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{2}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{53}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)+F_{58}\! \left(x \right)+F_{68}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{11}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{23}\! \left(x \right) F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{53}\! \left(x \right)+F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{62}\! \left(x \right)+F_{63}\! \left(x \right)+F_{64}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{15}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{63}\! \left(x \right) &= 0\\
F_{64}\! \left(x \right) &= F_{23}\! \left(x \right) F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{66}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{53}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{61}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{23}\! \left(x \right) F_{69}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{66}\! \left(x \right)+F_{70}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{56}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{72}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{68}\! \left(x \right)+F_{7}\! \left(x \right)+F_{73}\! \left(x \right)+F_{75}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{23}\! \left(x \right) F_{74}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{23}\! \left(x \right) F_{76}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{77}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{78}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{63}\! \left(x \right)+F_{64}\! \left(x \right)+F_{7}\! \left(x \right)+F_{79}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{23}\! \left(x \right) F_{80}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{82}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{64}\! \left(x \right)+F_{7}\! \left(x \right)+F_{83}\! \left(x \right)+F_{85}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{23}\! \left(x \right) F_{84}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{85}\! \left(x \right) &= 0\\
F_{86}\! \left(x \right) &= F_{23}\! \left(x \right) F_{87}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{88}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{6}\! \left(x \right)+F_{89}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{68}\! \left(x \right)+F_{7}\! \left(x \right)+F_{90}\! \left(x \right)+F_{92}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{23}\! \left(x \right) F_{91}\! \left(x \right)\\
F_{91}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{92}\! \left(x \right) &= F_{23}\! \left(x \right) F_{93}\! \left(x \right)\\
F_{93}\! \left(x \right) &= F_{77}\! \left(x \right)\\
\end{align*}\)