Av(1243, 1432, 2134, 2143, 2413)
Generating Function
\(\displaystyle \frac{x^{3}-2 x^{2}+3 x -1}{\left(3 x -1\right) \left(x^{2}-x +1\right)}\)
Counting Sequence
1, 1, 2, 6, 19, 58, 174, 521, 1562, 4686, 14059, 42178, 126534, 379601, 1138802, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(3 x -1\right) \left(x^{2}-x +1\right) F \! \left(x \right)-x^{3}+2 x^{2}-3 x +1 = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(n +3\right) = 3 a \! \left(n \right)-4 a \! \left(n +1\right)+4 a \! \left(n +2\right), \quad n \geq 4\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(n +3\right) = 3 a \! \left(n \right)-4 a \! \left(n +1\right)+4 a \! \left(n +2\right), \quad n \geq 4\)
Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0 \\ -\frac{i \left(\frac{1}{2}+\frac{i \sqrt{3}}{2}\right)^{-n} \sqrt{3}}{147}+\\\frac{i \left(\frac{1}{2}-\frac{i \sqrt{3}}{2}\right)^{-n} \sqrt{3}}{147}+\frac{5 \,3^{n}}{21}+\\\frac{3 \left(\frac{1}{2}+\frac{i \sqrt{3}}{2}\right)^{-n +1}}{49}+\frac{3 \left(\frac{1}{2}-\frac{i \sqrt{3}}{2}\right)^{-n +1}}{49}+\\\frac{9 \left(\frac{1}{2}+\frac{i \sqrt{3}}{2}\right)^{-n}}{49}+\frac{9 \left(\frac{1}{2}-\frac{i \sqrt{3}}{2}\right)^{-n}}{49} & \text{otherwise} \end{array}\right.\)
This specification was found using the strategy pack "Point Placements" and has 62 rules.
Found on January 18, 2022.Finding the specification took 1 seconds.
Copy 62 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_{20}\! \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_{11}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{15}\! \left(x \right) &= 0\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{13}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{22}\! \left(x \right)+F_{54}\! \left(x \right)\\
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_{33}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{4}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{15}\! \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_{17}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{11}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{28}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{36}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{22}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{2}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{37}\! \left(x \right)+F_{38}\! \left(x \right)+F_{48}\! \left(x \right)\\
F_{37}\! \left(x \right) &= 0\\
F_{38}\! \left(x \right) &= F_{39}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{43}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{4}\! \left(x \right) F_{40}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{44}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{45}\! \left(x \right)+F_{48}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{4}\! \left(x \right) F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{45}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{4}\! \left(x \right) F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{4}\! \left(x \right) F_{51}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{49}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{4}\! \left(x \right) F_{43}\! \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_{59}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{22}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{4}\! \left(x \right) F_{56}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{49}\! \left(x \right)+F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{45}\! \left(x \right)+F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{4}\! \left(x \right) F_{59}\! \left(x \right)\\
\end{align*}\)