Av(1234, 1243, 1342, 2134, 3214)
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Generating Function
\(\displaystyle \frac{x^{4}-x^{3}-2 x +1}{x^{6}-2 x^{3}+x^{2}-3 x +1}\)
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Counting Sequence
1, 1, 2, 6, 19, 55, 157, 453, 1310, 3785, 10932, 31576, 91209, 263462, 761019, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(x^{6}-2 x^{3}+x^{2}-3 x +1\right) F \! \left(x \right)-x^{4}+x^{3}+2 x -1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 55\)
\(\displaystyle a \! \left(n \right) = 2 a \! \left(n +3\right)-a \! \left(n +4\right)+3 a \! \left(n +5\right)-a \! \left(n +6\right), \quad n \geq 6\)
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Explicit Closed Form
\(\displaystyle -\frac{12496 \mathit{RootOf} \left(Z^{6}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =1\right)^{-n +4}}{477805}-\frac{12496 \mathit{RootOf} \left(Z^{6}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)^{-n +4}}{477805}-\frac{12496 \mathit{RootOf} \left(Z^{6}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)^{-n +4}}{477805}-\frac{12496 \mathit{RootOf} \left(Z^{6}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)^{-n +4}}{477805}-\frac{12496 \mathit{RootOf} \left(Z^{6}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =5\right)^{-n +4}}{477805}-\frac{12496 \mathit{RootOf} \left(Z^{6}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =6\right)^{-n +4}}{477805}-\frac{39891 \mathit{RootOf} \left(Z^{6}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =1\right)^{-n +3}}{477805}-\frac{39891 \mathit{RootOf} \left(Z^{6}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)^{-n +3}}{477805}-\frac{39891 \mathit{RootOf} \left(Z^{6}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)^{-n +3}}{477805}-\frac{39891 \mathit{RootOf} \left(Z^{6}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)^{-n +3}}{477805}-\frac{39891 \mathit{RootOf} \left(Z^{6}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =5\right)^{-n +3}}{477805}-\frac{39891 \mathit{RootOf} \left(Z^{6}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =6\right)^{-n +3}}{477805}+\frac{1261 \mathit{RootOf} \left(Z^{6}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =1\right)^{-n +2}}{477805}+\frac{1261 \mathit{RootOf} \left(Z^{6}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)^{-n +2}}{477805}+\frac{1261 \mathit{RootOf} \left(Z^{6}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)^{-n +2}}{477805}+\frac{1261 \mathit{RootOf} \left(Z^{6}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)^{-n +2}}{477805}+\frac{1261 \mathit{RootOf} \left(Z^{6}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =5\right)^{-n +2}}{477805}+\frac{1261 \mathit{RootOf} \left(Z^{6}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =6\right)^{-n +2}}{477805}+\frac{6686 \mathit{RootOf} \left(Z^{6}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =1\right)^{-n +1}}{95561}+\frac{6686 \mathit{RootOf} \left(Z^{6}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)^{-n +1}}{95561}+\frac{6686 \mathit{RootOf} \left(Z^{6}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)^{-n +1}}{95561}+\frac{6686 \mathit{RootOf} \left(Z^{6}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)^{-n +1}}{95561}+\frac{6686 \mathit{RootOf} \left(Z^{6}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =5\right)^{-n +1}}{95561}+\frac{6686 \mathit{RootOf} \left(Z^{6}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =6\right)^{-n +1}}{95561}+\frac{677 \mathit{RootOf} \left(Z^{6}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =1\right)^{-n -1}}{95561}+\frac{677 \mathit{RootOf} \left(Z^{6}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)^{-n -1}}{95561}+\frac{677 \mathit{RootOf} \left(Z^{6}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)^{-n -1}}{95561}+\frac{677 \mathit{RootOf} \left(Z^{6}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)^{-n -1}}{95561}+\frac{677 \mathit{RootOf} \left(Z^{6}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =5\right)^{-n -1}}{95561}+\frac{677 \mathit{RootOf} \left(Z^{6}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =6\right)^{-n -1}}{95561}+\frac{109502 \mathit{RootOf} \left(Z^{6}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =1\right)^{-n}}{477805}+\frac{109502 \mathit{RootOf} \left(Z^{6}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =2\right)^{-n}}{477805}+\frac{109502 \mathit{RootOf} \left(Z^{6}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =3\right)^{-n}}{477805}+\frac{109502 \mathit{RootOf} \left(Z^{6}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =4\right)^{-n}}{477805}+\frac{109502 \mathit{RootOf} \left(Z^{6}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =5\right)^{-n}}{477805}+\frac{109502 \mathit{RootOf} \left(Z^{6}-2 Z^{3}+Z^{2}-3 Z +1, \mathit{index} =6\right)^{-n}}{477805}\)
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This specification was found using the strategy pack "Point Placements" and has 52 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_{15}\! \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_{4}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{18}\! \left(x \right)+F_{38}\! \left(x \right)\\ F_{17}\! \left(x \right) &= 0\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{24}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{21}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{17}\! \left(x \right)+F_{25}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{31}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{30}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{31}\! \left(x \right) &= 2 F_{17}\! \left(x \right)+F_{32}\! \left(x \right)+F_{36}\! \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_{2}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{37}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{31}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{39}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{50}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{35}\! \left(x \right)+F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{4}\! \left(x \right) F_{43}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{47}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{45}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{17}\! \left(x \right)+F_{46}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{4}\! \left(x \right) F_{7}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{48}\! \left(x \right)\\ F_{48}\! \left(x \right) &= 2 F_{17}\! \left(x \right)+F_{32}\! \left(x \right)+F_{49}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{16}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{16}\! \left(x \right)+F_{51}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{32}\! \left(x \right)\\ \end{align*}\)