Av(1243, 1342, 1423, 1432, 2143, 3412)
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Generating Function
\(\displaystyle -\frac{x^{4}+x^{3}+2 x^{2}-3 x +1}{\left(x -1\right) \left(x^{2}-3 x +1\right)}\)
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Counting Sequence
1, 1, 2, 6, 18, 50, 134, 354, 930, 2438, 6386, 16722, 43782, 114626, 300098, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(x -1\right) \left(x^{2}-3 x +1\right) F \! \left(x \right)+x^{4}+x^{3}+2 x^{2}-3 x +1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 18\)
\(\displaystyle a \! \left(n +2\right) = -a \! \left(n \right)+3 a \! \left(n +1\right)+2, \quad n \geq 5\)
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Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ \frac{\left(-8 \sqrt{5}+20\right) \left(\frac{3}{2}-\frac{\sqrt{5}}{2}\right)^{-n}}{5}-2+\frac{\left(8 \sqrt{5}+20\right) \left(\frac{3}{2}+\frac{\sqrt{5}}{2}\right)^{-n}}{5} & \text{otherwise} \end{array}\right.\)
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This specification was found using the strategy pack "Point Placements" and has 23 rules.

Found on July 23, 2021.

Finding the specification took 5 seconds.

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\(\begin{align*} F_{0}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{1}\! \left(x \right) &= 1\\ F_{2}\! \left(x \right) &= F_{3}\! \left(x \right)\\ F_{3}\! \left(x \right) &= F_{17}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{4}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{5}\! \left(x \right)\\ F_{5}\! \left(x \right) &= F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{17}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{14}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{0}\! \left(x \right)+F_{10}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right) F_{17}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{18}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{14}\! \left(x \right) F_{17}\! \left(x \right)\\ F_{17}\! \left(x \right) &= x\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{14}\! \left(x \right) F_{17}\! \left(x \right) F_{20}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{15}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{14}\! \left(x \right) F_{17}\! \left(x \right) F_{20}\! \left(x \right)\\ \end{align*}\)