Av(1234, 1243, 2143, 2413, 3214)
Generating Function
\(\displaystyle \frac{x^{3}+x^{2}+x -1}{2 x^{5}+3 x^{4}+2 x^{3}+x^{2}+2 x -1}\)
Counting Sequence
1, 1, 2, 6, 19, 53, 145, 403, 1126, 3142, 8757, 24407, 68039, 189677, 528762, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(2 x^{5}+3 x^{4}+2 x^{3}+x^{2}+2 x -1\right) F \! \left(x \right)-x^{3}-x^{2}-x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(n +5\right) = 2 a \! \left(n \right)+3 a \! \left(n +1\right)+2 a \! \left(n +2\right)+a \! \left(n +3\right)+2 a \! \left(n +4\right), \quad n \geq 5\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(n +5\right) = 2 a \! \left(n \right)+3 a \! \left(n +1\right)+2 a \! \left(n +2\right)+a \! \left(n +3\right)+2 a \! \left(n +4\right), \quad n \geq 5\)
Explicit Closed Form
\(\displaystyle -\frac{1351 \mathit{RootOf} \left(2 Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =1\right)^{-n +3}}{23377}-\frac{1351 \mathit{RootOf} \left(2 Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =2\right)^{-n +3}}{23377}-\frac{1351 \mathit{RootOf} \left(2 Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =3\right)^{-n +3}}{23377}-\frac{1351 \mathit{RootOf} \left(2 Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =4\right)^{-n +3}}{23377}-\frac{1351 \mathit{RootOf} \left(2 Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =5\right)^{-n +3}}{23377}-\frac{271 \mathit{RootOf} \left(2 Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =1\right)^{-n +2}}{46754}-\frac{271 \mathit{RootOf} \left(2 Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =2\right)^{-n +2}}{46754}-\frac{271 \mathit{RootOf} \left(2 Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =3\right)^{-n +2}}{46754}-\frac{271 \mathit{RootOf} \left(2 Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =4\right)^{-n +2}}{46754}-\frac{271 \mathit{RootOf} \left(2 Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =5\right)^{-n +2}}{46754}+\frac{4045 \mathit{RootOf} \left(2 Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =1\right)^{-n +1}}{46754}+\frac{4045 \mathit{RootOf} \left(2 Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =2\right)^{-n +1}}{46754}+\frac{4045 \mathit{RootOf} \left(2 Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =3\right)^{-n +1}}{46754}+\frac{4045 \mathit{RootOf} \left(2 Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =4\right)^{-n +1}}{46754}+\frac{4045 \mathit{RootOf} \left(2 Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =5\right)^{-n +1}}{46754}+\frac{1163 \mathit{RootOf} \left(2 Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =1\right)^{-n -1}}{46754}+\frac{1163 \mathit{RootOf} \left(2 Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =2\right)^{-n -1}}{46754}+\frac{1163 \mathit{RootOf} \left(2 Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =3\right)^{-n -1}}{46754}+\frac{1163 \mathit{RootOf} \left(2 Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =4\right)^{-n -1}}{46754}+\frac{1163 \mathit{RootOf} \left(2 Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =5\right)^{-n -1}}{46754}+\frac{4955 \mathit{RootOf} \left(2 Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =1\right)^{-n}}{23377}+\frac{4955 \mathit{RootOf} \left(2 Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =2\right)^{-n}}{23377}+\frac{4955 \mathit{RootOf} \left(2 Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =3\right)^{-n}}{23377}+\frac{4955 \mathit{RootOf} \left(2 Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =4\right)^{-n}}{23377}+\frac{4955 \mathit{RootOf} \left(2 Z^{5}+3 Z^{4}+2 Z^{3}+Z^{2}+2 Z -1, \mathit{index} =5\right)^{-n}}{23377}\)
This specification was found using the strategy pack "Point Placements" and has 61 rules.
Found on January 18, 2022.Finding the specification took 1 seconds.
Copy 61 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_{19}\! \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_{14}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{13}\! \left(x \right) &= 0\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{4}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{4}\! \left(x \right) F_{7}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{21}\! \left(x \right)+F_{50}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{26}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{10}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{14}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{11}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{31}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{2}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{34}\! \left(x \right) &= 2 F_{13}\! \left(x \right)+F_{35}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{32}\! \left(x \right)\\
F_{38}\! \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_{41}\! \left(x \right) &= F_{4}\! \left(x \right) F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= 2 F_{13}\! \left(x \right)+F_{39}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{4}\! \left(x \right) F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{48}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{32}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{39}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{4}\! \left(x \right) F_{43}\! \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_{59}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{33}\! \left(x \right)+F_{54}\! \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_{57}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= 2 F_{13}\! \left(x \right)+F_{35}\! \left(x \right)+F_{39}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= 2 F_{13}\! \left(x \right)+F_{39}\! \left(x \right)+F_{45}\! \left(x \right)\\
\end{align*}\)