Av(1342, 1432, 2413, 3124, 3214)
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Generating Function
\(\displaystyle -\frac{\left(x -1\right) \left(x^{2}+x -1\right)}{x^{5}+3 x^{4}-x^{2}+3 x -1}\)
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Counting Sequence
1, 1, 2, 6, 19, 55, 153, 424, 1182, 3306, 9250, 25869, 72327, 202212, 565365, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(x^{5}+3 x^{4}-x^{2}+3 x -1\right) F \! \left(x \right)+\left(x -1\right) \left(x^{2}+x -1\right) = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(n +1\right) = -\frac{a \! \left(n \right)}{3}+\frac{a \! \left(n +3\right)}{3}-a \! \left(n +4\right)+\frac{a \! \left(n +5\right)}{3}, \quad n \geq 5\)
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Explicit Closed Form
\(\displaystyle -\frac{1186 \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =1\right)^{-n +3}}{44719}-\frac{1186 \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =2\right)^{-n +3}}{44719}-\frac{1186 \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)^{-n +3}}{44719}-\frac{1186 \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)^{-n +3}}{44719}-\frac{1186 \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =5\right)^{-n +3}}{44719}-\frac{2697 \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)^{-n +2}}{44719}-\frac{2697 \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)^{-n +2}}{44719}-\frac{2697 \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =5\right)^{-n +2}}{44719}-\frac{2697 \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =1\right)^{-n +2}}{44719}-\frac{2697 \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =2\right)^{-n +2}}{44719}+\frac{1694 \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =1\right)^{-n +1}}{44719}+\frac{1694 \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =2\right)^{-n +1}}{44719}+\frac{1694 \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)^{-n +1}}{44719}+\frac{1694 \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)^{-n +1}}{44719}+\frac{1694 \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =5\right)^{-n +1}}{44719}+\frac{2205 \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =1\right)^{-n -1}}{44719}+\frac{2205 \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =2\right)^{-n -1}}{44719}+\frac{2205 \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)^{-n -1}}{44719}+\frac{2205 \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)^{-n -1}}{44719}+\frac{2205 \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =5\right)^{-n -1}}{44719}+\frac{7799 \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =1\right)^{-n}}{44719}+\frac{7799 \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =2\right)^{-n}}{44719}+\frac{7799 \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =3\right)^{-n}}{44719}+\frac{7799 \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =4\right)^{-n}}{44719}+\frac{7799 \mathit{RootOf} \left(Z^{5}+3 Z^{4}-Z^{2}+3 Z -1, \mathit{index} =5\right)^{-n}}{44719}\)
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This specification was found using the strategy pack "Point Placements" and has 45 rules.

Found on January 18, 2022.

Finding the specification took 0 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_{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_{6}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{14}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{11}\! \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_{7}\! \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_{41}\! \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_{26}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{11}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{38}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \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_{2}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{35}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{34}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{32}\! \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_{27}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{4}\! \left(x \right) F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{32}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{4}\! \left(x \right) F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{43}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{44}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{36}\! \left(x \right)\\ \end{align*}\)