Av(1243, 1324, 1342, 4132, 4231)
Generating Function
\(\displaystyle -\frac{x^{8}-3 x^{7}-x^{6}+13 x^{5}-34 x^{4}+42 x^{3}-26 x^{2}+8 x -1}{\left(2 x -1\right)^{3} \left(x -1\right)^{3}}\)
Counting Sequence
1, 1, 2, 6, 19, 56, 152, 388, 949, 2253, 5236, 11978, 27071, 60595, 134566, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(2 x -1\right)^{3} \left(x -1\right)^{3} F \! \left(x \right)+x^{8}-3 x^{7}-x^{6}+13 x^{5}-34 x^{4}+42 x^{3}-26 x^{2}+8 x -1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 56\)
\(\displaystyle a \! \left(6\right) = 152\)
\(\displaystyle a \! \left(7\right) = 388\)
\(\displaystyle a \! \left(8\right) = 949\)
\(\displaystyle a \! \left(n +3\right) = \frac{n^{2}}{2}+8 a \! \left(n \right)-12 a \! \left(n +1\right)+6 a \! \left(n +2\right)-\frac{7 n}{2}+2, \quad n \geq 9\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 56\)
\(\displaystyle a \! \left(6\right) = 152\)
\(\displaystyle a \! \left(7\right) = 388\)
\(\displaystyle a \! \left(8\right) = 949\)
\(\displaystyle a \! \left(n +3\right) = \frac{n^{2}}{2}+8 a \! \left(n \right)-12 a \! \left(n +1\right)+6 a \! \left(n +2\right)-\frac{7 n}{2}+2, \quad n \geq 9\)
Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ \frac{\left(n^{2}+25 n -20\right) 2^{n}}{64}-\frac{n^{2}}{2}+\frac{n}{2}+1 & \text{otherwise} \end{array}\right.\)
This specification was found using the strategy pack "Point Placements" and has 75 rules.
Found on July 23, 2021.Finding the specification took 6 seconds.
Copy 75 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_{19}\! \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_{13}\! \left(x \right)\\
F_{10}\! \left(x \right) &= x\\
F_{11}\! \left(x \right) &= 2 F_{12}\! \left(x \right)+F_{1}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{10}\! \left(x \right) F_{11}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{15}\! \left(x \right) &= F_{10}\! \left(x \right) F_{13}\! \left(x \right) F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{17}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{10}\! \left(x \right) F_{16}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{10}\! \left(x \right) F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{13}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{27}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{1}\! \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_{23}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{16}\! \left(x \right) F_{17}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{10}\! \left(x \right) F_{11}\! \left(x \right) F_{17}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{62}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{16}\! \left(x \right) F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{36}\! \left(x \right)+F_{44}\! \left(x \right)\\
F_{35}\! \left(x \right) &= 0\\
F_{36}\! \left(x \right) &= F_{10}\! \left(x \right) F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{10}\! \left(x \right) F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{36}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{10}\! \left(x \right) F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{10}\! \left(x \right) F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{48}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{10}\! \left(x \right) F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{48}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{10}\! \left(x \right) F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{54}\! \left(x \right)+F_{59}\! \left(x \right)+F_{61}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{10}\! \left(x \right) F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{10}\! \left(x \right) F_{38}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{54}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{10}\! \left(x \right) F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{10}\! \left(x \right) F_{46}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{27}\! \left(x \right) F_{63}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{64}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{36}\! \left(x \right)+F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{10}\! \left(x \right) F_{66}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{68}\! \left(x \right)+F_{71}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{69}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{10}\! \left(x \right) F_{70}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{68}\! \left(x \right)\\
F_{71}\! \left(x \right) &= 2 F_{35}\! \left(x \right)+F_{72}\! \left(x \right)+F_{74}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{10}\! \left(x \right) F_{73}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{71}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{10}\! \left(x \right) F_{67}\! \left(x \right)\\
\end{align*}\)