Av(1342, 1432, 2341, 3241)
Generating Function
\(\displaystyle \frac{x^{3}-x^{2}+3 x -1}{2 x^{3}-3 x^{2}+4 x -1}\)
Counting Sequence
1, 1, 2, 6, 20, 66, 216, 706, 2308, 7546, 24672, 80666, 263740, 862306, 2819336, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(2 x^{3}-3 x^{2}+4 x -1\right) F \! \left(x \right)-x^{3}+x^{2}-3 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(n +3\right) = 2 a \! \left(n \right)-3 a \! \left(n +1\right)+4 a \! \left(n +2\right), \quad n \geq 4\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(n +3\right) = 2 a \! \left(n \right)-3 a \! \left(n +1\right)+4 a \! \left(n +2\right), \quad n \geq 4\)
Explicit Closed Form
\(\displaystyle \frac{4 \left(\underset{\alpha =\mathit{RootOf} \left(2 Z^{3}-3 Z^{2}+4 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +1}\right)}{19}+\frac{\left(\underset{\alpha =\mathit{RootOf} \left(2 Z^{3}-3 Z^{2}+4 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n}\right)}{38}+\frac{\left(\underset{\alpha =\mathit{RootOf} \left(2 Z^{3}-3 Z^{2}+4 Z -1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n -1}\right)}{38}+\left(\left\{\begin{array}{cc}\frac{1}{2} & n =0 \\ 0 & \text{otherwise} \end{array}\right.\right)\)
This specification was found using the strategy pack "Point Placements" and has 69 rules.
Found on January 18, 2022.Finding the specification took 1 seconds.
Copy 69 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_{10}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\
F_{7}\! \left(x \right) &= F_{10}\! \left(x \right) F_{8}\! \left(x \right)\\
F_{8}\! \left(x \right) &= F_{5}\! \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) &= x\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{13}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{12}\! \left(x \right) &= 0\\
F_{13}\! \left(x \right) &= F_{10}\! \left(x \right) F_{6}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{10}\! \left(x \right) F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{9}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{10}\! \left(x \right) F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{10}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{10}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{25}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{10}\! \left(x \right) F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{10}\! \left(x \right) F_{17}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{31}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{10}\! \left(x \right) F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{14}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{10}\! \left(x \right) F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{38}\! \left(x \right) &= 2 F_{12}\! \left(x \right)+F_{39}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{10}\! \left(x \right) F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= 2 F_{12}\! \left(x \right)+F_{42}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{10}\! \left(x \right) F_{30}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{10}\! \left(x \right) F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{10}\! \left(x \right) F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{29}\! \left(x \right)+F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{10}\! \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_{10}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= x^{2}\\
F_{55}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{10}\! \left(x \right) F_{28}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{42}\! \left(x \right)+F_{59}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{10}\! \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_{11}\! \left(x \right)+F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{63}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{10}\! \left(x \right) F_{11}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{66}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{10}\! \left(x \right) F_{41}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{10}\! \left(x \right) F_{68}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{48}\! \left(x \right)\\
\end{align*}\)