Av(12453, 12543, 14253, 14523, 15243, 15423, 21453, 21543, 24153, 24513, 25143, 25413, 41253, 41523, 42153, 42513, 45123, 45213, 51243, 51423, 52143, 52413, 54123, 54213)
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 Left" and has 79 rules.
Finding the specification took 204 seconds.
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Copy 79 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_{15}\! \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_{7}\! \left(x \right)+F_{72}\! \left(x \right)+F_{8}\! \left(x \right)\\
F_{7}\! \left(x \right) &= 0\\
F_{8}\! \left(x \right) &= F_{15}\! \left(x \right) F_{9}\! \left(x \right)\\
F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{68}\! \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_{66}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{12}\! \left(x \right) &= F_{13}\! \left(x \right) F_{15}\! \left(x \right)\\
F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{16}\! \left(x \right)\\
F_{15}\! \left(x \right) &= x\\
F_{16}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{17}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{15}\! \left(x \right) F_{2}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{22}\! \left(x \right)+F_{62}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{15}\! \left(x \right) F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{15}\! \left(x \right) F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{61}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{28}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{15}\! \left(x \right) F_{27}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{28}\! \left(x \right) &= x^{2}\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{32}\! \left(x \right)+F_{39}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{15}\! \left(x \right) F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{15}\! \left(x \right) F_{33}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{60}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{36}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{36}\! \left(x \right) &= x^{2}\\
F_{37}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{38}\! \left(x \right)+F_{39}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{15}\! \left(x \right) F_{16}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{15}\! \left(x \right) F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{42}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{15}\! \left(x \right) F_{2}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{15}\! \left(x \right) F_{43}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{45}\! \left(x \right)\\
F_{44}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{45}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{48}\! \left(x \right)+F_{58}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{11}\! \left(x \right) F_{15}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{15}\! \left(x \right) F_{49}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)+F_{57}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)+F_{54}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{52}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{15}\! \left(x \right) F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{15}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{38}\! \left(x \right)+F_{55}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{15}\! \left(x \right) F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{40}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{54}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{15}\! \left(x \right) F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{40}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{37}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{29}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{15}\! \left(x \right) F_{63}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{64}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{15}\! \left(x \right) F_{65}\! \left(x \right)\\
F_{65}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{66}\! \left(x \right) &= F_{15}\! \left(x \right) F_{67}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{2}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{69}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{22}\! \left(x \right)+F_{7}\! \left(x \right)+F_{70}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{15}\! \left(x \right) F_{71}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{63}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{15}\! \left(x \right) F_{73}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{74}\! \left(x \right)+F_{75}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{63}\! \left(x \right)+F_{76}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{48}\! \left(x \right)+F_{58}\! \left(x \right)+F_{7}\! \left(x \right)+F_{77}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{15}\! \left(x \right) F_{78}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{11}\! \left(x \right)\\
\end{align*}\)