Av(1234, 1243, 1342, 2413, 3142)
Generating Function
\(\displaystyle \frac{\left(x^{3}-2 x^{2}+3 x -1\right) \left(x -1\right)^{3}}{3 x^{6}-12 x^{5}+25 x^{4}-28 x^{3}+19 x^{2}-7 x +1}\)
Counting Sequence
1, 1, 2, 6, 19, 57, 166, 482, 1407, 4124, 12108, 35567, 104500, 307089, 902566, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(3 x^{6}-12 x^{5}+25 x^{4}-28 x^{3}+19 x^{2}-7 x +1\right) F \! \left(x \right)-\left(x^{3}-2 x^{2}+3 x -1\right) \left(x -1\right)^{3} = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 57\)
\(\displaystyle a \! \left(6\right) = 166\)
\(\displaystyle a \! \left(n +6\right) = -3 a \! \left(n \right)+12 a \! \left(n +1\right)-25 a \! \left(n +2\right)+28 a \! \left(n +3\right)-19 a \! \left(n +4\right)+7 a \! \left(n +5\right), \quad n \geq 7\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 57\)
\(\displaystyle a \! \left(6\right) = 166\)
\(\displaystyle a \! \left(n +6\right) = -3 a \! \left(n \right)+12 a \! \left(n +1\right)-25 a \! \left(n +2\right)+28 a \! \left(n +3\right)-19 a \! \left(n +4\right)+7 a \! \left(n +5\right), \quad n \geq 7\)
Explicit Closed Form
\(\displaystyle -\frac{21699 \mathit{RootOf} \left(3 Z^{6}-12 Z^{5}+25 Z^{4}-28 Z^{3}+19 Z^{2}-7 Z +1, \mathit{index} =1\right)^{-n +4}}{38023}-\frac{21699 \mathit{RootOf} \left(3 Z^{6}-12 Z^{5}+25 Z^{4}-28 Z^{3}+19 Z^{2}-7 Z +1, \mathit{index} =2\right)^{-n +4}}{38023}-\frac{21699 \mathit{RootOf} \left(3 Z^{6}-12 Z^{5}+25 Z^{4}-28 Z^{3}+19 Z^{2}-7 Z +1, \mathit{index} =3\right)^{-n +4}}{38023}-\frac{21699 \mathit{RootOf} \left(3 Z^{6}-12 Z^{5}+25 Z^{4}-28 Z^{3}+19 Z^{2}-7 Z +1, \mathit{index} =4\right)^{-n +4}}{38023}-\frac{21699 \mathit{RootOf} \left(3 Z^{6}-12 Z^{5}+25 Z^{4}-28 Z^{3}+19 Z^{2}-7 Z +1, \mathit{index} =5\right)^{-n +4}}{38023}-\frac{21699 \mathit{RootOf} \left(3 Z^{6}-12 Z^{5}+25 Z^{4}-28 Z^{3}+19 Z^{2}-7 Z +1, \mathit{index} =6\right)^{-n +4}}{38023}+\frac{77949 \mathit{RootOf} \left(3 Z^{6}-12 Z^{5}+25 Z^{4}-28 Z^{3}+19 Z^{2}-7 Z +1, \mathit{index} =1\right)^{-n +3}}{38023}+\frac{77949 \mathit{RootOf} \left(3 Z^{6}-12 Z^{5}+25 Z^{4}-28 Z^{3}+19 Z^{2}-7 Z +1, \mathit{index} =2\right)^{-n +3}}{38023}+\frac{77949 \mathit{RootOf} \left(3 Z^{6}-12 Z^{5}+25 Z^{4}-28 Z^{3}+19 Z^{2}-7 Z +1, \mathit{index} =3\right)^{-n +3}}{38023}+\frac{77949 \mathit{RootOf} \left(3 Z^{6}-12 Z^{5}+25 Z^{4}-28 Z^{3}+19 Z^{2}-7 Z +1, \mathit{index} =4\right)^{-n +3}}{38023}+\frac{77949 \mathit{RootOf} \left(3 Z^{6}-12 Z^{5}+25 Z^{4}-28 Z^{3}+19 Z^{2}-7 Z +1, \mathit{index} =5\right)^{-n +3}}{38023}+\frac{77949 \mathit{RootOf} \left(3 Z^{6}-12 Z^{5}+25 Z^{4}-28 Z^{3}+19 Z^{2}-7 Z +1, \mathit{index} =6\right)^{-n +3}}{38023}-\frac{141553 \mathit{RootOf} \left(3 Z^{6}-12 Z^{5}+25 Z^{4}-28 Z^{3}+19 Z^{2}-7 Z +1, \mathit{index} =1\right)^{-n +2}}{38023}-\frac{141553 \mathit{RootOf} \left(3 Z^{6}-12 Z^{5}+25 Z^{4}-28 Z^{3}+19 Z^{2}-7 Z +1, \mathit{index} =2\right)^{-n +2}}{38023}-\frac{141553 \mathit{RootOf} \left(3 Z^{6}-12 Z^{5}+25 Z^{4}-28 Z^{3}+19 Z^{2}-7 Z +1, \mathit{index} =3\right)^{-n +2}}{38023}-\frac{141553 \mathit{RootOf} \left(3 Z^{6}-12 Z^{5}+25 Z^{4}-28 Z^{3}+19 Z^{2}-7 Z +1, \mathit{index} =4\right)^{-n +2}}{38023}-\frac{141553 \mathit{RootOf} \left(3 Z^{6}-12 Z^{5}+25 Z^{4}-28 Z^{3}+19 Z^{2}-7 Z +1, \mathit{index} =5\right)^{-n +2}}{38023}-\frac{141553 \mathit{RootOf} \left(3 Z^{6}-12 Z^{5}+25 Z^{4}-28 Z^{3}+19 Z^{2}-7 Z +1, \mathit{index} =6\right)^{-n +2}}{38023}+\frac{123205 \mathit{RootOf} \left(3 Z^{6}-12 Z^{5}+25 Z^{4}-28 Z^{3}+19 Z^{2}-7 Z +1, \mathit{index} =1\right)^{-n +1}}{38023}+\frac{123205 \mathit{RootOf} \left(3 Z^{6}-12 Z^{5}+25 Z^{4}-28 Z^{3}+19 Z^{2}-7 Z +1, \mathit{index} =2\right)^{-n +1}}{38023}+\frac{123205 \mathit{RootOf} \left(3 Z^{6}-12 Z^{5}+25 Z^{4}-28 Z^{3}+19 Z^{2}-7 Z +1, \mathit{index} =3\right)^{-n +1}}{38023}+\frac{123205 \mathit{RootOf} \left(3 Z^{6}-12 Z^{5}+25 Z^{4}-28 Z^{3}+19 Z^{2}-7 Z +1, \mathit{index} =4\right)^{-n +1}}{38023}+\frac{123205 \mathit{RootOf} \left(3 Z^{6}-12 Z^{5}+25 Z^{4}-28 Z^{3}+19 Z^{2}-7 Z +1, \mathit{index} =5\right)^{-n +1}}{38023}+\frac{123205 \mathit{RootOf} \left(3 Z^{6}-12 Z^{5}+25 Z^{4}-28 Z^{3}+19 Z^{2}-7 Z +1, \mathit{index} =6\right)^{-n +1}}{38023}+\frac{13060 \mathit{RootOf} \left(3 Z^{6}-12 Z^{5}+25 Z^{4}-28 Z^{3}+19 Z^{2}-7 Z +1, \mathit{index} =1\right)^{-n -1}}{38023}+\frac{13060 \mathit{RootOf} \left(3 Z^{6}-12 Z^{5}+25 Z^{4}-28 Z^{3}+19 Z^{2}-7 Z +1, \mathit{index} =2\right)^{-n -1}}{38023}+\frac{13060 \mathit{RootOf} \left(3 Z^{6}-12 Z^{5}+25 Z^{4}-28 Z^{3}+19 Z^{2}-7 Z +1, \mathit{index} =3\right)^{-n -1}}{38023}+\frac{13060 \mathit{RootOf} \left(3 Z^{6}-12 Z^{5}+25 Z^{4}-28 Z^{3}+19 Z^{2}-7 Z +1, \mathit{index} =4\right)^{-n -1}}{38023}+\frac{13060 \mathit{RootOf} \left(3 Z^{6}-12 Z^{5}+25 Z^{4}-28 Z^{3}+19 Z^{2}-7 Z +1, \mathit{index} =5\right)^{-n -1}}{38023}+\frac{13060 \mathit{RootOf} \left(3 Z^{6}-12 Z^{5}+25 Z^{4}-28 Z^{3}+19 Z^{2}-7 Z +1, \mathit{index} =6\right)^{-n -1}}{38023}-\frac{57183 \mathit{RootOf} \left(3 Z^{6}-12 Z^{5}+25 Z^{4}-28 Z^{3}+19 Z^{2}-7 Z +1, \mathit{index} =1\right)^{-n}}{38023}-\frac{57183 \mathit{RootOf} \left(3 Z^{6}-12 Z^{5}+25 Z^{4}-28 Z^{3}+19 Z^{2}-7 Z +1, \mathit{index} =2\right)^{-n}}{38023}-\frac{57183 \mathit{RootOf} \left(3 Z^{6}-12 Z^{5}+25 Z^{4}-28 Z^{3}+19 Z^{2}-7 Z +1, \mathit{index} =3\right)^{-n}}{38023}-\frac{57183 \mathit{RootOf} \left(3 Z^{6}-12 Z^{5}+25 Z^{4}-28 Z^{3}+19 Z^{2}-7 Z +1, \mathit{index} =4\right)^{-n}}{38023}-\frac{57183 \mathit{RootOf} \left(3 Z^{6}-12 Z^{5}+25 Z^{4}-28 Z^{3}+19 Z^{2}-7 Z +1, \mathit{index} =5\right)^{-n}}{38023}-\frac{57183 \mathit{RootOf} \left(3 Z^{6}-12 Z^{5}+25 Z^{4}-28 Z^{3}+19 Z^{2}-7 Z +1, \mathit{index} =6\right)^{-n}}{38023}+\left(\left\{\begin{array}{cc}\frac{1}{3} & n =0 \\ 0 & \text{otherwise} \end{array}\right.\right)\)
This specification was found using the strategy pack "Point Placements" and has 65 rules.
Found on January 18, 2022.Finding the specification took 1 seconds.
Copy 65 equations to clipboard:
\(\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_{17}\! \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_{11}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{16}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{4}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{12}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{20}\! \left(x \right)+F_{50}\! \left(x \right)\\
F_{19}\! \left(x \right) &= 0\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{13}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= 2 F_{19}\! \left(x \right)+F_{26}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{23}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{25}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{34}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{38}\! \left(x \right) &= 2 F_{19}\! \left(x \right)+F_{30}\! \left(x \right)+F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{4}\! \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_{12}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{38}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{2}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{4}\! \left(x \right) F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{33}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{45}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{4}\! \left(x \right) F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{33}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{4}\! \left(x \right) F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{43}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{54}\! \left(x \right)+F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{4}\! \left(x \right) F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{63}\! \left(x \right)+F_{64}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{58}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{60}\! \left(x \right)\\
\end{align*}\)