Av(1342, 1423, 1432, 2431, 3214)
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Generating Function
\(\displaystyle -\frac{\left(x -1\right) \left(x^{6}+2 x^{5}+x^{4}+2 x^{3}+2 x^{2}-3 x +1\right)}{\left(x^{2}+x -1\right) \left(3 x^{3}-5 x^{2}+4 x -1\right)}\)
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Counting Sequence
1, 1, 2, 6, 19, 55, 147, 376, 944, 2353, 5846, 14493, 35869, 88659, 218948, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(x^{2}+x -1\right) \left(3 x^{3}-5 x^{2}+4 x -1\right) F \! \left(x \right)+\left(x -1\right) \left(x^{6}+2 x^{5}+x^{4}+2 x^{3}+2 x^{2}-3 x +1\right) = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 55\)
\(\displaystyle a \! \left(6\right) = 147\)
\(\displaystyle a \! \left(7\right) = 376\)
\(\displaystyle a \! \left(n +5\right) = -3 a \! \left(n \right)+2 a \! \left(n +1\right)+4 a \! \left(n +2\right)-8 a \! \left(n +3\right)+5 a \! \left(n +4\right), \quad n \geq 8\)
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Explicit Closed Form
\(\displaystyle -\frac{707 \left(\underset{\alpha =\mathit{RootOf} \left(3 Z^{5}-2 Z^{4}-4 Z^{3}+8 Z^{2}-5 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n +3}\right)}{155}+\frac{67 \left(\underset{\alpha =\mathit{RootOf} \left(3 Z^{5}-2 Z^{4}-4 Z^{3}+8 Z^{2}-5 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{2-n}\right)}{465}+\frac{8896 \left(\underset{\alpha =\mathit{RootOf} \left(3 Z^{5}-2 Z^{4}-4 Z^{3}+8 Z^{2}-5 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{1-n}\right)}{1395}-\frac{10781 \left(\underset{\alpha =\mathit{RootOf} \left(3 Z^{5}-2 Z^{4}-4 Z^{3}+8 Z^{2}-5 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n}\right)}{1395}+\frac{3473 \left(\underset{\alpha =\mathit{RootOf} \left(3 Z^{5}-2 Z^{4}-4 Z^{3}+8 Z^{2}-5 Z +1\right)}{\textcolor{gray}{\sum}}\! \alpha^{-n -1}\right)}{1395}-\frac{\left(\left\{\begin{array}{cc}\frac{13}{9} & n =0 \\ \frac{5}{3} & n =1 \\ 1 & n =2 \\ 0 & \text{otherwise} \end{array}\right.\right)}{3}\)
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This specification was found using the strategy pack "Point Placements" and has 51 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_{13}\! \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_{4}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{4}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{16}\! \left(x \right)+F_{41}\! \left(x \right)\\ F_{15}\! \left(x \right) &= 0\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{24}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{20}\! \left(x \right)+F_{22}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{18}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{10}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{25}\! \left(x \right)\\ F_{29}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{30}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{24}\! \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_{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_{37}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{4}\! \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_{37}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{4}\! \left(x \right) F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{43}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{44}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{4}\! \left(x \right) F_{46}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{48}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{4}\! \left(x \right) F_{50}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{47}\! \left(x \right)\\ \end{align*}\)