Av(1243, 2143, 2413, 4132, 4213)
Generating Function
\(\displaystyle -\frac{3 x^{4}-9 x^{3}+12 x^{2}-6 x +1}{\left(2 x^{3}-4 x^{2}+4 x -1\right) \left(x^{2}-3 x +1\right)}\)
Counting Sequence
1, 1, 2, 6, 19, 59, 180, 543, 1625, 4832, 14291, 42073, 123376, 360559, 1050589, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(2 x^{3}-4 x^{2}+4 x -1\right) \left(x^{2}-3 x +1\right) F \! \left(x \right)+3 x^{4}-9 x^{3}+12 x^{2}-6 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(n +5\right) = 2 a \! \left(n \right)-10 a \! \left(n +1\right)+18 a \! \left(n +2\right)-17 a \! \left(n +3\right)+7 a \! \left(n +4\right), \quad n \geq 5\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(n +5\right) = 2 a \! \left(n \right)-10 a \! \left(n +1\right)+18 a \! \left(n +2\right)-17 a \! \left(n +3\right)+7 a \! \left(n +4\right), \quad n \geq 5\)
Explicit Closed Form
\(\displaystyle \frac{344 \left(\underset{\alpha =\mathit{RootOf} \left(2 Z^{5}-10 Z^{4}+18 Z^{3}-17 Z^{2}+7 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +3}\right)}{55}-\frac{1589 \left(\underset{\alpha =\mathit{RootOf} \left(2 Z^{5}-10 Z^{4}+18 Z^{3}-17 Z^{2}+7 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +2}\right)}{55}+\frac{500 \left(\underset{\alpha =\mathit{RootOf} \left(2 Z^{5}-10 Z^{4}+18 Z^{3}-17 Z^{2}+7 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +1}\right)}{11}-\frac{1979 \left(\underset{\alpha =\mathit{RootOf} \left(2 Z^{5}-10 Z^{4}+18 Z^{3}-17 Z^{2}+7 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n}\right)}{55}+\frac{463 \left(\underset{\alpha =\mathit{RootOf} \left(2 Z^{5}-10 Z^{4}+18 Z^{3}-17 Z^{2}+7 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n -1}\right)}{55}\)
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_{31}\! \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_{13}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{10}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{16}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{17}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{22}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{21}\! \left(x \right) &= 0\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{24}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{27}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{13}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{33}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{39}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{26}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{33}\! \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_{14}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{43}\! \left(x \right) &= 2 F_{21}\! \left(x \right)+F_{44}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{4}\! \left(x \right) F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{44}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{4}\! \left(x \right) F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{4}\! \left(x \right) F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{4}\! \left(x \right) F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{52}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{4}\! \left(x \right) F_{55}\! \left(x \right)\\
F_{58}\! \left(x \right) &= 2 F_{21}\! \left(x \right)+F_{59}\! \left(x \right)+F_{64}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{4}\! \left(x \right) F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{58}\! \left(x \right)+F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{63}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{4}\! \left(x \right) F_{61}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{4}\! \left(x \right) F_{51}\! \left(x \right)\\
\end{align*}\)