Av(12345, 12354, 12435, 12453, 12534, 12543, 13245, 13254, 13452, 13542, 14253, 14352, 21345, 21354, 21435, 21453, 21534, 21543, 23145, 23154, 24153, 31245, 31254, 32145, 32154)
Generating Function
\(\displaystyle -\frac{\left(x -1\right) \left(x^{4}-3 x^{3}-5 x^{2}-2 x +1\right)}{x^{6}-5 x^{5}+x^{4}+5 x^{3}-x^{2}-4 x +1}\)
Counting Sequence
1, 1, 2, 6, 24, 95, 376, 1482, 5833, 22953, 90310, 355331, 1398070, 5500791, 21643201, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{6}-5 x^{5}+x^{4}+5 x^{3}-x^{2}-4 x +1\right) F \! \left(x \right)+\left(x -1\right) \left(x^{4}-3 x^{3}-5 x^{2}-2 x +1\right) = 0\)
Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 95\)
\(\displaystyle a{\left(n + 6 \right)} = - a{\left(n \right)} + 5 a{\left(n + 1 \right)} - a{\left(n + 2 \right)} - 5 a{\left(n + 3 \right)} + a{\left(n + 4 \right)} + 4 a{\left(n + 5 \right)}, \quad n \geq 6\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a(4) = 24\)
\(\displaystyle a(5) = 95\)
\(\displaystyle a{\left(n + 6 \right)} = - a{\left(n \right)} + 5 a{\left(n + 1 \right)} - a{\left(n + 2 \right)} - 5 a{\left(n + 3 \right)} + a{\left(n + 4 \right)} + 4 a{\left(n + 5 \right)}, \quad n \geq 6\)
Explicit Closed Form
\(\displaystyle \frac{2482733 \mathit{RootOf} \left(Z^{6}-5 Z^{5}+Z^{4}+5 Z^{3}-Z^{2}-4 Z +1, \mathit{index} =1\right)^{-n +4}}{37031606}+\frac{2482733 \mathit{RootOf} \left(Z^{6}-5 Z^{5}+Z^{4}+5 Z^{3}-Z^{2}-4 Z +1, \mathit{index} =2\right)^{-n +4}}{37031606}+\frac{2482733 \mathit{RootOf} \left(Z^{6}-5 Z^{5}+Z^{4}+5 Z^{3}-Z^{2}-4 Z +1, \mathit{index} =3\right)^{-n +4}}{37031606}+\frac{2482733 \mathit{RootOf} \left(Z^{6}-5 Z^{5}+Z^{4}+5 Z^{3}-Z^{2}-4 Z +1, \mathit{index} =4\right)^{-n +4}}{37031606}+\frac{2482733 \mathit{RootOf} \left(Z^{6}-5 Z^{5}+Z^{4}+5 Z^{3}-Z^{2}-4 Z +1, \mathit{index} =5\right)^{-n +4}}{37031606}+\frac{2482733 \mathit{RootOf} \left(Z^{6}-5 Z^{5}+Z^{4}+5 Z^{3}-Z^{2}-4 Z +1, \mathit{index} =6\right)^{-n +4}}{37031606}-\frac{5758797 \mathit{RootOf} \left(Z^{6}-5 Z^{5}+Z^{4}+5 Z^{3}-Z^{2}-4 Z +1, \mathit{index} =1\right)^{-n +3}}{18515803}-\frac{5758797 \mathit{RootOf} \left(Z^{6}-5 Z^{5}+Z^{4}+5 Z^{3}-Z^{2}-4 Z +1, \mathit{index} =2\right)^{-n +3}}{18515803}-\frac{5758797 \mathit{RootOf} \left(Z^{6}-5 Z^{5}+Z^{4}+5 Z^{3}-Z^{2}-4 Z +1, \mathit{index} =3\right)^{-n +3}}{18515803}-\frac{5758797 \mathit{RootOf} \left(Z^{6}-5 Z^{5}+Z^{4}+5 Z^{3}-Z^{2}-4 Z +1, \mathit{index} =4\right)^{-n +3}}{18515803}-\frac{5758797 \mathit{RootOf} \left(Z^{6}-5 Z^{5}+Z^{4}+5 Z^{3}-Z^{2}-4 Z +1, \mathit{index} =5\right)^{-n +3}}{18515803}-\frac{5758797 \mathit{RootOf} \left(Z^{6}-5 Z^{5}+Z^{4}+5 Z^{3}-Z^{2}-4 Z +1, \mathit{index} =6\right)^{-n +3}}{18515803}-\frac{2505073 \mathit{RootOf} \left(Z^{6}-5 Z^{5}+Z^{4}+5 Z^{3}-Z^{2}-4 Z +1, \mathit{index} =1\right)^{-n +2}}{37031606}-\frac{2505073 \mathit{RootOf} \left(Z^{6}-5 Z^{5}+Z^{4}+5 Z^{3}-Z^{2}-4 Z +1, \mathit{index} =2\right)^{-n +2}}{37031606}-\frac{2505073 \mathit{RootOf} \left(Z^{6}-5 Z^{5}+Z^{4}+5 Z^{3}-Z^{2}-4 Z +1, \mathit{index} =3\right)^{-n +2}}{37031606}-\frac{2505073 \mathit{RootOf} \left(Z^{6}-5 Z^{5}+Z^{4}+5 Z^{3}-Z^{2}-4 Z +1, \mathit{index} =4\right)^{-n +2}}{37031606}-\frac{2505073 \mathit{RootOf} \left(Z^{6}-5 Z^{5}+Z^{4}+5 Z^{3}-Z^{2}-4 Z +1, \mathit{index} =5\right)^{-n +2}}{37031606}-\frac{2505073 \mathit{RootOf} \left(Z^{6}-5 Z^{5}+Z^{4}+5 Z^{3}-Z^{2}-4 Z +1, \mathit{index} =6\right)^{-n +2}}{37031606}+\frac{6784155 \mathit{RootOf} \left(Z^{6}-5 Z^{5}+Z^{4}+5 Z^{3}-Z^{2}-4 Z +1, \mathit{index} =1\right)^{-n +1}}{18515803}+\frac{6784155 \mathit{RootOf} \left(Z^{6}-5 Z^{5}+Z^{4}+5 Z^{3}-Z^{2}-4 Z +1, \mathit{index} =2\right)^{-n +1}}{18515803}+\frac{6784155 \mathit{RootOf} \left(Z^{6}-5 Z^{5}+Z^{4}+5 Z^{3}-Z^{2}-4 Z +1, \mathit{index} =3\right)^{-n +1}}{18515803}+\frac{6784155 \mathit{RootOf} \left(Z^{6}-5 Z^{5}+Z^{4}+5 Z^{3}-Z^{2}-4 Z +1, \mathit{index} =4\right)^{-n +1}}{18515803}+\frac{6784155 \mathit{RootOf} \left(Z^{6}-5 Z^{5}+Z^{4}+5 Z^{3}-Z^{2}-4 Z +1, \mathit{index} =5\right)^{-n +1}}{18515803}+\frac{6784155 \mathit{RootOf} \left(Z^{6}-5 Z^{5}+Z^{4}+5 Z^{3}-Z^{2}-4 Z +1, \mathit{index} =6\right)^{-n +1}}{18515803}-\frac{2397045 \mathit{RootOf} \left(Z^{6}-5 Z^{5}+Z^{4}+5 Z^{3}-Z^{2}-4 Z +1, \mathit{index} =1\right)^{-n -1}}{37031606}-\frac{2397045 \mathit{RootOf} \left(Z^{6}-5 Z^{5}+Z^{4}+5 Z^{3}-Z^{2}-4 Z +1, \mathit{index} =2\right)^{-n -1}}{37031606}-\frac{2397045 \mathit{RootOf} \left(Z^{6}-5 Z^{5}+Z^{4}+5 Z^{3}-Z^{2}-4 Z +1, \mathit{index} =3\right)^{-n -1}}{37031606}-\frac{2397045 \mathit{RootOf} \left(Z^{6}-5 Z^{5}+Z^{4}+5 Z^{3}-Z^{2}-4 Z +1, \mathit{index} =4\right)^{-n -1}}{37031606}-\frac{2397045 \mathit{RootOf} \left(Z^{6}-5 Z^{5}+Z^{4}+5 Z^{3}-Z^{2}-4 Z +1, \mathit{index} =5\right)^{-n -1}}{37031606}-\frac{2397045 \mathit{RootOf} \left(Z^{6}-5 Z^{5}+Z^{4}+5 Z^{3}-Z^{2}-4 Z +1, \mathit{index} =6\right)^{-n -1}}{37031606}+\frac{10084737 \mathit{RootOf} \left(Z^{6}-5 Z^{5}+Z^{4}+5 Z^{3}-Z^{2}-4 Z +1, \mathit{index} =1\right)^{-n}}{37031606}+\frac{10084737 \mathit{RootOf} \left(Z^{6}-5 Z^{5}+Z^{4}+5 Z^{3}-Z^{2}-4 Z +1, \mathit{index} =2\right)^{-n}}{37031606}+\frac{10084737 \mathit{RootOf} \left(Z^{6}-5 Z^{5}+Z^{4}+5 Z^{3}-Z^{2}-4 Z +1, \mathit{index} =3\right)^{-n}}{37031606}+\frac{10084737 \mathit{RootOf} \left(Z^{6}-5 Z^{5}+Z^{4}+5 Z^{3}-Z^{2}-4 Z +1, \mathit{index} =4\right)^{-n}}{37031606}+\frac{10084737 \mathit{RootOf} \left(Z^{6}-5 Z^{5}+Z^{4}+5 Z^{3}-Z^{2}-4 Z +1, \mathit{index} =5\right)^{-n}}{37031606}+\frac{10084737 \mathit{RootOf} \left(Z^{6}-5 Z^{5}+Z^{4}+5 Z^{3}-Z^{2}-4 Z +1, \mathit{index} =6\right)^{-n}}{37031606}\)
This specification was found using the strategy pack "Regular Insertion Encoding Bottom" and has 105 rules.
Finding the specification took 159 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_{15}\! \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_{38}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{15}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right) F_{15}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{16}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= x\\
F_{16}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{18}\! \left(x \right) &= 0\\
F_{19}\! \left(x \right) &= F_{15}\! \left(x \right) F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{23}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{15}\! \left(x \right) F_{6}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{15}\! \left(x \right) F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{22}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{29}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{29}\! \left(x \right) &= x^{2}\\
F_{30}\! \left(x \right) &= F_{15}\! \left(x \right) F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{33}\! \left(x \right)+F_{34}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{15}\! \left(x \right) F_{22}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{15}\! \left(x \right) F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{22}\! \left(x \right)\\
F_{37}\! \left(x \right) &= 0\\
F_{38}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{39}\! \left(x \right)+F_{97}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{15}\! \left(x \right) F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{93}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{44}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{15}\! \left(x \right) F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{69}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{48}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{15}\! \left(x \right) F_{2}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{15}\! \left(x \right) F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{54}\! \left(x \right)+F_{55}\! \left(x \right)+F_{82}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{15}\! \left(x \right) F_{38}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{15}\! \left(x \right) F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)+F_{77}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{54}\! \left(x \right)+F_{59}\! \left(x \right)+F_{76}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{15}\! \left(x \right) F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{61}\! \left(x \right)+F_{64}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{62}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{48}\! \left(x \right)\\
F_{63}\! \left(x \right) &= 2 F_{18}\! \left(x \right)+F_{54}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{65}\! \left(x \right)+F_{70}\! \left(x \right)\\
F_{65}\! \left(x \right) &= 2 F_{18}\! \left(x \right)+F_{66}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{15}\! \left(x \right) F_{47}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{15}\! \left(x \right) F_{68}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{69}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{62}\! \left(x \right)\\
F_{70}\! \left(x \right) &= 2 F_{18}\! \left(x \right)+F_{71}\! \left(x \right)+F_{72}\! \left(x \right)+F_{75}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{15}\! \left(x \right) F_{58}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{15}\! \left(x \right) F_{73}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{74}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{63}\! \left(x \right)\\
F_{75}\! \left(x \right) &= 0\\
F_{76}\! \left(x \right) &= 0\\
F_{77}\! \left(x \right) &= F_{78}\! \left(x \right)+F_{80}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{66}\! \left(x \right)+F_{67}\! \left(x \right)+F_{79}\! \left(x \right)\\
F_{79}\! \left(x \right) &= 0\\
F_{80}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{71}\! \left(x \right)+F_{72}\! \left(x \right)+F_{75}\! \left(x \right)+F_{81}\! \left(x \right)\\
F_{81}\! \left(x \right) &= 0\\
F_{82}\! \left(x \right) &= F_{15}\! \left(x \right) F_{83}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{84}\! \left(x \right)+F_{85}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{78}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{86}\! \left(x \right)\\
F_{86}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{87}\! \left(x \right)+F_{88}\! \left(x \right)+F_{91}\! \left(x \right)+F_{92}\! \left(x \right)\\
F_{87}\! \left(x \right) &= F_{15}\! \left(x \right) F_{53}\! \left(x \right)\\
F_{88}\! \left(x \right) &= F_{15}\! \left(x \right) F_{89}\! \left(x \right)\\
F_{89}\! \left(x \right) &= F_{90}\! \left(x \right)\\
F_{90}\! \left(x \right) &= F_{63}\! \left(x \right)\\
F_{91}\! \left(x \right) &= 0\\
F_{92}\! \left(x \right) &= 0\\
F_{93}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{59}\! \left(x \right)+F_{76}\! \left(x \right)+F_{94}\! \left(x \right)\\
F_{94}\! \left(x \right) &= F_{15}\! \left(x \right) F_{95}\! \left(x \right)\\
F_{95}\! \left(x \right) &= F_{74}\! \left(x \right)+F_{96}\! \left(x \right)\\
F_{96}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{97}\! \left(x \right) &= F_{15}\! \left(x \right) F_{98}\! \left(x \right)\\
F_{98}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{99}\! \left(x \right)\\
F_{99}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{100}\! \left(x \right) &= F_{101}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{101}\! \left(x \right) &= F_{102}\! \left(x \right)+F_{18}\! \left(x \right)+F_{55}\! \left(x \right)+F_{82}\! \left(x \right)\\
F_{102}\! \left(x \right) &= F_{103}\! \left(x \right) F_{15}\! \left(x \right)\\
F_{103}\! \left(x \right) &= F_{104}\! \left(x \right)+F_{90}\! \left(x \right)\\
F_{104}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{53}\! \left(x \right)\\
\end{align*}\)