Av(1234, 1243, 1342, 3124, 3142)
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Generating Function
\(\displaystyle -\frac{\left(2 x -1\right) \left(x -1\right)^{2}}{x^{4}-6 x^{3}+8 x^{2}-5 x +1}\)
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Counting Sequence
1, 1, 2, 6, 19, 58, 172, 504, 1473, 4307, 12603, 36893, 108010, 316217, 925760, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(x^{4}-6 x^{3}+8 x^{2}-5 x +1\right) F \! \left(x \right)+\left(2 x -1\right) \left(x -1\right)^{2} = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(n +4\right) = -a \! \left(n \right)+6 a \! \left(n +1\right)-8 a \! \left(n +2\right)+5 a \! \left(n +3\right), \quad n \geq 4\)
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Explicit Closed Form
\(\displaystyle \frac{46 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +2}\right)}{563}-\frac{215 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +1}\right)}{563}+\frac{242 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n}\right)}{563}-\frac{7 \left(\underset{\alpha =\mathit{RootOf} \left(Z^{4}-6 Z^{3}+8 Z^{2}-5 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n -1}\right)}{563}\)
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This specification was found using the strategy pack "Point Placements" and has 39 rules.

Found on January 18, 2022.

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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_{4}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{4}\! \left(x \right) &= x\\ F_{5}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{6}\! \left(x \right)\\ F_{6}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{7}\! \left(x \right) &= F_{8}\! \left(x \right)\\ F_{8}\! \left(x \right) &= F_{4}\! \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_{1}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{16}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{4}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{12}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{20}\! \left(x \right)+F_{24}\! \left(x \right)\\ F_{19}\! \left(x \right) &= 0\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{7}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{18}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{33}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{28}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{31}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{4}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{27}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{2}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{34}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{36}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{38}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{27}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{34}\! \left(x \right)\\ \end{align*}\)