Av(1243, 1342, 1432, 3412, 4231)
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Generating Function
\(\displaystyle \frac{x^{6}+4 x^{5}-4 x^{4}+8 x^{3}-10 x^{2}+5 x -1}{\left(2 x -1\right) \left(x -1\right)^{4}}\)
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Counting Sequence
1, 1, 2, 6, 19, 51, 119, 254, 515, 1017, 1987, 3876, 7583, 14903, 29423, ...
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Implicit Equation for the Generating Function
\(\displaystyle -\left(2 x -1\right) \left(x -1\right)^{4} F \! \left(x \right)+x^{6}+4 x^{5}-4 x^{4}+8 x^{3}-10 x^{2}+5 x -1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 51\)
\(\displaystyle a \! \left(6\right) = 119\)
\(\displaystyle a \! \left(n +1\right) = -\frac{n^{3}}{2}+5 n^{2}+2 a \! \left(n \right)-\frac{21 n}{2}+7, \quad n \geq 7\)
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Explicit Closed Form
\(\displaystyle \left\{\begin{array}{cc}1 & n =0\text{ or } n =1 \\ -5+\frac{n^{3}}{2}-\frac{7 n^{2}}{2}+5 n +\frac{7 \,2^{n}}{4} & \text{otherwise} \end{array}\right.\)
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This specification was found using the strategy pack "Point Placements" and has 34 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_{14}\! \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_{14}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{9}\! \left(x \right)\\ F_{9}\! \left(x \right) &= F_{10}\! \left(x \right)+F_{17}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right) F_{15}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{12}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right) F_{9}\! \left(x \right)\\ F_{14}\! \left(x \right) &= x\\ F_{15}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{16}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{14}\! \left(x \right) F_{15}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right) F_{21}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{19}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{14}\! \left(x \right) F_{18}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{14} \left(x \right)^{2} F_{15} \left(x \right)^{2}\\ F_{23}\! \left(x \right) &= F_{15}\! \left(x \right) F_{18}\! \left(x \right) F_{19}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{15}\! \left(x \right) F_{19}\! \left(x \right) F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{19}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{18}\! \left(x \right) F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{14}\! \left(x \right) F_{19}\! \left(x \right) F_{30}\! \left(x \right) F_{31}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{14}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{14}\! \left(x \right) F_{15}\! \left(x \right) F_{31}\! \left(x \right)\\ \end{align*}\)