Av(13524, 13542, 15324, 15342, 31524, 31542, 35124, 35142, 51324, 51342, 53124, 53142)
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Generating Function
\(\displaystyle \frac{2 x^{2}-5 x +1}{6 x^{2}-6 x +1}\)
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Counting Sequence
1, 1, 2, 6, 24, 108, 504, 2376, 11232, 53136, 251424, 1189728, 5629824, 26640576, 126064512, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(6 x^{2}-6 x +1\right) F \! \left(x \right)-2 x^{2}+5 x -1 = 0\)
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Recurrence
\(\displaystyle a(0) = 1\)
\(\displaystyle a(1) = 1\)
\(\displaystyle a(2) = 2\)
\(\displaystyle a{\left(n + 2 \right)} = - 6 a{\left(n \right)} + 6 a{\left(n + 1 \right)}, \quad n \geq 3\)
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Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0 \\ \frac{\left(2-\sqrt{3}\right) \left(\frac{1}{2}-\frac{\sqrt{3}}{6}\right)^{-n}}{6}+\frac{\left(\frac{1}{2}+\frac{\sqrt{3}}{6}\right)^{-n} \left(2+\sqrt{3}\right)}{6} & \text{otherwise} \end{array}\right.\)
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This specification was found using the strategy pack "Regular Insertion Encoding Left" and has 16 rules.

Finding the specification took 43 seconds.

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\(\begin{align*} F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{1}\! \left(x \right) &= 1\\ F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{12}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{7}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{7}\! \left(x \right) &= 0\\ F_{8}\! \left(x \right) &= F_{12}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{11}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{6}\! \left(x \right)\\ F_{12}\! \left(x \right) &= x\\ F_{13}\! \left(x \right) &= F_{12}\! \left(x \right) F_{14}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{6}\! \left(x \right)\\ \end{align*}\)