Av(12345, 12354, 12435, 12453, 12534, 12543, 13245, 13254, 13425, 13452, 13524, 13542, 14235, 14253, 14325, 14352, 14523, 14532, 15234, 15243, 15324, 15342, 15423, 15432, 21345, 21354, 21435, 21453, 21534, 21543)
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Generating Function
\(\displaystyle \frac{3 x^{3}+4 x^{2}+2 x -1}{3 x^{2}+3 x -1}\)
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Counting Sequence
1, 1, 2, 6, 24, 90, 342, 1296, 4914, 18630, 70632, 267786, 1015254, 3849120, 14593122, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(-3 x^{2}-3 x +1\right) F \! \left(x \right)+3 x^{3}+4 x^{2}+2 x -1 = 0\)
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Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a(3) = 6\)
\(\displaystyle a{\left(n + 2 \right)} = 3 a{\left(n \right)} + 3 a{\left(n + 1 \right)}, \quad n \geq 4\)
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Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ \frac{\left(\sqrt{21}+7\right) \left(-\frac{1}{2}-\frac{\sqrt{21}}{6}\right)^{-n}}{21}-\frac{\left(-\frac{1}{2}+\frac{\sqrt{21}}{6}\right)^{-n} \left(\sqrt{21}-7\right)}{21} & \text{otherwise} \end{array}\right.\)
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This specification was found using the strategy pack "Regular Insertion Encoding Bottom" and has 72 rules.

Finding the specification took 389 seconds.

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\(\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_{16}\! \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_{33}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{16}\! \left(x \right) F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{19}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{15}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{17}\! \left(x \right)\\ F_{16}\! \left(x \right) &= x\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{11}\! \left(x \right) F_{16}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{21}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{20}\! \left(x \right) &= 0\\ F_{21}\! \left(x \right) &= F_{16}\! \left(x \right) F_{22}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{24}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{16}\! \left(x \right) F_{6}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{16}\! \left(x \right) F_{19}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{16}\! \left(x \right) F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{25}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{34}\! \left(x \right)+F_{68}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{16}\! \left(x \right) F_{35}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)+F_{37}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{33}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{53}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{39}\! \left(x \right)+F_{47}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{16}\! \left(x \right) F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{42}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{38}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{45}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{16}\! \left(x \right) F_{2}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{16}\! \left(x \right) F_{38}\! \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_{50}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{43}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{51}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{16}\! \left(x \right) F_{33}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{54}\! \left(x \right)+F_{60}\! \left(x \right)+F_{64}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{16}\! \left(x \right) F_{55}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)+F_{57}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{53}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{51}\! \left(x \right)+F_{58}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{59}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{16}\! \left(x \right) F_{53}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{16}\! \left(x \right) F_{61}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{62}\! \left(x \right)+F_{63}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{51}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{58}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{16}\! \left(x \right) F_{65}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{66}\! \left(x \right)+F_{67}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{45}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{58}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{16}\! \left(x \right) F_{69}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{70}\! \left(x \right)+F_{71}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{38}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{53}\! \left(x \right)\\ \end{align*}\)