Av(12354, 12453, 13254, 13452, 14253, 14352, 21354, 21453, 23154, 23451, 24153, 24351, 31254, 31452, 32154, 32451, 34152, 34251, 41253, 41352, 42153, 42351, 43152, 43251)
Generating Function
\(\displaystyle \frac{2 x^{3}+2 x^{2}+3 x -1}{4 x -1}\)
Counting Sequence
1, 1, 2, 6, 24, 96, 384, 1536, 6144, 24576, 98304, 393216, 1572864, 6291456, 25165824, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(-4 x +1\right) F \! \left(x \right)+2 x^{3}+2 x^{2}+3 x -1 = 0\)
Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a{\left(n + 1 \right)} = 4 a{\left(n \right)}, \quad n \geq 4\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a{\left(n + 1 \right)} = 4 a{\left(n \right)}, \quad n \geq 4\)
Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ 2 & n =2 \\ \frac{3 \,4^{n}}{32} & \text{otherwise} \end{array}\right.\)
This specification was found using the strategy pack "Regular Insertion Encoding Bottom" and has 70 rules.
Finding the specification took 97 seconds.
Copy 70 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_{14}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\
F_{5}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{6}\! \left(x \right)\\
F_{6}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{7}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{7}\! \left(x \right) &= 0\\
F_{8}\! \left(x \right) &= F_{14}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{39}\! \left(x \right)\\
F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{15}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right) F_{14}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{14}\! \left(x \right) &= x\\
F_{15}\! \left(x \right) &= F_{14}\! \left(x \right) F_{16}\! \left(x \right)\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{20}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{19}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{14}\! \left(x \right) F_{2}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{23}\! \left(x \right)+F_{25}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{11}\! \left(x \right) F_{14}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{14}\! \left(x \right) F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{14}\! \left(x \right) F_{26}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{33}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{30}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{29}\! \left(x \right) &= x^{2}\\
F_{30}\! \left(x \right) &= F_{14}\! \left(x \right) F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{14}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{34}\! \left(x \right)+F_{35}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{14}\! \left(x \right) F_{18}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{14}\! \left(x \right) F_{36}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{33}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{14}\! \left(x \right) F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{51}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{14}\! \left(x \right) F_{45}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{48}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{14}\! \left(x \right) F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)+F_{55}\! \left(x \right)+F_{60}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{14}\! \left(x \right) F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{14}\! \left(x \right) F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{58}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{14}\! \left(x \right) F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{14}\! \left(x \right) F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{62}\! \left(x \right)+F_{69}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{63}\! \left(x \right)+F_{66}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{64}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{14}\! \left(x \right) F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{14}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{60}\! \left(x \right)+F_{67}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{14}\! \left(x \right) F_{68}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{18}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{66}\! \left(x \right)\\
\end{align*}\)