Av(1234, 1342, 2314, 2413, 3142)
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Generating Function
\(\displaystyle -\frac{\left(x -1\right)^{4}}{x^{5}-3 x^{4}+9 x^{3}-9 x^{2}+5 x -1}\)
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Counting Sequence
1, 1, 2, 6, 19, 57, 163, 457, 1280, 3602, 10171, 28749, 81241, 229477, 648046, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(x^{5}-3 x^{4}+9 x^{3}-9 x^{2}+5 x -1\right) F \! \left(x \right)+\left(x -1\right)^{4} = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(n +5\right) = a \! \left(n \right)-3 a \! \left(n +1\right)+9 a \! \left(n +2\right)-9 a \! \left(n +3\right)+5 a \! \left(n +4\right), \quad n \geq 5\)
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Explicit Closed Form
\(\displaystyle \frac{19 \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =1\right)^{-n +3}}{2563}+\frac{19 \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =2\right)^{-n +3}}{2563}+\frac{19 \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)^{-n +3}}{2563}+\frac{19 \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =4\right)^{-n +3}}{2563}+\frac{19 \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =5\right)^{-n +3}}{2563}-\frac{9 \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =1\right)^{-n +2}}{233}-\frac{9 \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =2\right)^{-n +2}}{233}-\frac{9 \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)^{-n +2}}{233}-\frac{9 \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =4\right)^{-n +2}}{233}-\frac{9 \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =5\right)^{-n +2}}{233}+\frac{375 \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =1\right)^{-n +1}}{5126}+\frac{375 \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =2\right)^{-n +1}}{5126}+\frac{375 \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)^{-n +1}}{5126}+\frac{375 \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =4\right)^{-n +1}}{5126}+\frac{375 \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =5\right)^{-n +1}}{5126}+\frac{471 \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =1\right)^{-n -1}}{5126}+\frac{471 \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =2\right)^{-n -1}}{5126}+\frac{471 \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)^{-n -1}}{5126}+\frac{471 \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =4\right)^{-n -1}}{5126}+\frac{471 \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =5\right)^{-n -1}}{5126}+\frac{89 \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =1\right)^{-n}}{2563}+\frac{89 \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =2\right)^{-n}}{2563}+\frac{89 \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =3\right)^{-n}}{2563}+\frac{89 \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =4\right)^{-n}}{2563}+\frac{89 \mathit{RootOf} \left(Z^{5}-3 Z^{4}+9 Z^{3}-9 Z^{2}+5 Z -1, \mathit{index} =5\right)^{-n}}{2563}\)
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This specification was found using the strategy pack "Point Placements" and has 44 rules.

Found on January 18, 2022.

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_{4}\! \left(x \right) F_{5}\! \left(x \right)\\ F_{4}\! \left(x \right) &= x\\ F_{5}\! \left(x \right) &= F_{19}\! \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_{13}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{11}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{10}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{18}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{11}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{14}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{20}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{22}\! \left(x \right)+F_{40}\! \left(x \right)\\ F_{21}\! \left(x \right) &= 0\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{26}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= 2 F_{21}\! \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_{25}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{27}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{33}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{38}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{37}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{35}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{32}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{4}\! \left(x \right) F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{43}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{40}\! \left(x \right)\\ \end{align*}\)