Av(1243, 1324, 1342, 1423, 1432, 2143, 2413, 2431, 3142, 3412, 4132)
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Generating Function
\(\displaystyle \frac{x^{5}-3 x^{4}+x^{3}-2 x +1}{x^{2}-3 x +1}\)
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Counting Sequence
1, 1, 2, 6, 13, 34, 89, 233, 610, 1597, 4181, 10946, 28657, 75025, 196418, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(-x^{2}+3 x -1\right) F \! \left(x \right)+x^{5}-3 x^{4}+x^{3}-2 x +1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 13\)
\(\displaystyle a \! \left(5\right) = 34\)
\(\displaystyle a \! \left(n +2\right) = -a \! \left(n \right)+3 a \! \left(n +1\right), \quad n \geq 6\)
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Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}2 & n =0 \\ \frac{\left(5-\sqrt{5}\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{10}+\frac{\left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n} \left(5+\sqrt{5}\right)}{10} & \text{otherwise} \end{array}\right.\)
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This specification was found using the strategy pack "Point Placements" and has 20 rules.

Found on July 23, 2021.

Finding the specification took 1 seconds.

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\(\begin{align*} F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{1}\! \left(x \right) &= 1\\ F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{13}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{13}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{14}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{12}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{11}\! \left(x \right) F_{13}\! \left(x \right)\\ F_{13}\! \left(x \right) &= x\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{11} \left(x \right)^{2} F_{13}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{13}\! \left(x \right) F_{18}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{19}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{11}\! \left(x \right) F_{9}\! \left(x \right)\\ \end{align*}\)