Av(1243, 2314, 3142, 3214)
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Generating Function
\(\displaystyle \frac{\left(x^{3}-2 x^{2}+3 x -1\right)^{2}}{3 x^{6}-9 x^{5}+20 x^{4}-24 x^{3}+18 x^{2}-7 x +1}\)
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Counting Sequence
1, 1, 2, 6, 20, 65, 206, 647, 2029, 6363, 19952, 62545, 196016, 614210, 1924402, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(3 x^{6}-9 x^{5}+20 x^{4}-24 x^{3}+18 x^{2}-7 x +1\right) F \! \left(x \right)-\left(x^{3}-2 x^{2}+3 x -1\right)^{2} = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 65\)
\(\displaystyle a \! \left(6\right) = 206\)
\(\displaystyle a \! \left(n +6\right) = -3 a \! \left(n \right)+9 a \! \left(n +1\right)-20 a \! \left(n +2\right)+24 a \! \left(n +3\right)-18 a \! \left(n +4\right)+7 a \! \left(n +5\right), \quad n \geq 7\)
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Explicit Closed Form
\(\displaystyle -\frac{128805 \mathit{RootOf} \left(3 Z^{6}-9 Z^{5}+20 Z^{4}-24 Z^{3}+18 Z^{2}-7 Z +1, \mathit{index} =1\right)^{-n +4}}{194194}-\frac{128805 \mathit{RootOf} \left(3 Z^{6}-9 Z^{5}+20 Z^{4}-24 Z^{3}+18 Z^{2}-7 Z +1, \mathit{index} =2\right)^{-n +4}}{194194}-\frac{128805 \mathit{RootOf} \left(3 Z^{6}-9 Z^{5}+20 Z^{4}-24 Z^{3}+18 Z^{2}-7 Z +1, \mathit{index} =3\right)^{-n +4}}{194194}-\frac{128805 \mathit{RootOf} \left(3 Z^{6}-9 Z^{5}+20 Z^{4}-24 Z^{3}+18 Z^{2}-7 Z +1, \mathit{index} =4\right)^{-n +4}}{194194}-\frac{128805 \mathit{RootOf} \left(3 Z^{6}-9 Z^{5}+20 Z^{4}-24 Z^{3}+18 Z^{2}-7 Z +1, \mathit{index} =5\right)^{-n +4}}{194194}-\frac{128805 \mathit{RootOf} \left(3 Z^{6}-9 Z^{5}+20 Z^{4}-24 Z^{3}+18 Z^{2}-7 Z +1, \mathit{index} =6\right)^{-n +4}}{194194}+\frac{332463 \mathit{RootOf} \left(3 Z^{6}-9 Z^{5}+20 Z^{4}-24 Z^{3}+18 Z^{2}-7 Z +1, \mathit{index} =1\right)^{-n +3}}{194194}+\frac{332463 \mathit{RootOf} \left(3 Z^{6}-9 Z^{5}+20 Z^{4}-24 Z^{3}+18 Z^{2}-7 Z +1, \mathit{index} =2\right)^{-n +3}}{194194}+\frac{332463 \mathit{RootOf} \left(3 Z^{6}-9 Z^{5}+20 Z^{4}-24 Z^{3}+18 Z^{2}-7 Z +1, \mathit{index} =3\right)^{-n +3}}{194194}+\frac{332463 \mathit{RootOf} \left(3 Z^{6}-9 Z^{5}+20 Z^{4}-24 Z^{3}+18 Z^{2}-7 Z +1, \mathit{index} =4\right)^{-n +3}}{194194}+\frac{332463 \mathit{RootOf} \left(3 Z^{6}-9 Z^{5}+20 Z^{4}-24 Z^{3}+18 Z^{2}-7 Z +1, \mathit{index} =5\right)^{-n +3}}{194194}+\frac{332463 \mathit{RootOf} \left(3 Z^{6}-9 Z^{5}+20 Z^{4}-24 Z^{3}+18 Z^{2}-7 Z +1, \mathit{index} =6\right)^{-n +3}}{194194}-\frac{99195 \mathit{RootOf} \left(3 Z^{6}-9 Z^{5}+20 Z^{4}-24 Z^{3}+18 Z^{2}-7 Z +1, \mathit{index} =1\right)^{-n +2}}{27742}-\frac{99195 \mathit{RootOf} \left(3 Z^{6}-9 Z^{5}+20 Z^{4}-24 Z^{3}+18 Z^{2}-7 Z +1, \mathit{index} =2\right)^{-n +2}}{27742}-\frac{99195 \mathit{RootOf} \left(3 Z^{6}-9 Z^{5}+20 Z^{4}-24 Z^{3}+18 Z^{2}-7 Z +1, \mathit{index} =3\right)^{-n +2}}{27742}-\frac{99195 \mathit{RootOf} \left(3 Z^{6}-9 Z^{5}+20 Z^{4}-24 Z^{3}+18 Z^{2}-7 Z +1, \mathit{index} =4\right)^{-n +2}}{27742}-\frac{99195 \mathit{RootOf} \left(3 Z^{6}-9 Z^{5}+20 Z^{4}-24 Z^{3}+18 Z^{2}-7 Z +1, \mathit{index} =5\right)^{-n +2}}{27742}-\frac{99195 \mathit{RootOf} \left(3 Z^{6}-9 Z^{5}+20 Z^{4}-24 Z^{3}+18 Z^{2}-7 Z +1, \mathit{index} =6\right)^{-n +2}}{27742}+\frac{690667 \mathit{RootOf} \left(3 Z^{6}-9 Z^{5}+20 Z^{4}-24 Z^{3}+18 Z^{2}-7 Z +1, \mathit{index} =1\right)^{-n +1}}{194194}+\frac{690667 \mathit{RootOf} \left(3 Z^{6}-9 Z^{5}+20 Z^{4}-24 Z^{3}+18 Z^{2}-7 Z +1, \mathit{index} =2\right)^{-n +1}}{194194}+\frac{690667 \mathit{RootOf} \left(3 Z^{6}-9 Z^{5}+20 Z^{4}-24 Z^{3}+18 Z^{2}-7 Z +1, \mathit{index} =3\right)^{-n +1}}{194194}+\frac{690667 \mathit{RootOf} \left(3 Z^{6}-9 Z^{5}+20 Z^{4}-24 Z^{3}+18 Z^{2}-7 Z +1, \mathit{index} =4\right)^{-n +1}}{194194}+\frac{690667 \mathit{RootOf} \left(3 Z^{6}-9 Z^{5}+20 Z^{4}-24 Z^{3}+18 Z^{2}-7 Z +1, \mathit{index} =5\right)^{-n +1}}{194194}+\frac{690667 \mathit{RootOf} \left(3 Z^{6}-9 Z^{5}+20 Z^{4}-24 Z^{3}+18 Z^{2}-7 Z +1, \mathit{index} =6\right)^{-n +1}}{194194}+\frac{43107 \mathit{RootOf} \left(3 Z^{6}-9 Z^{5}+20 Z^{4}-24 Z^{3}+18 Z^{2}-7 Z +1, \mathit{index} =1\right)^{-n -1}}{97097}+\frac{43107 \mathit{RootOf} \left(3 Z^{6}-9 Z^{5}+20 Z^{4}-24 Z^{3}+18 Z^{2}-7 Z +1, \mathit{index} =2\right)^{-n -1}}{97097}+\frac{43107 \mathit{RootOf} \left(3 Z^{6}-9 Z^{5}+20 Z^{4}-24 Z^{3}+18 Z^{2}-7 Z +1, \mathit{index} =3\right)^{-n -1}}{97097}+\frac{43107 \mathit{RootOf} \left(3 Z^{6}-9 Z^{5}+20 Z^{4}-24 Z^{3}+18 Z^{2}-7 Z +1, \mathit{index} =4\right)^{-n -1}}{97097}+\frac{43107 \mathit{RootOf} \left(3 Z^{6}-9 Z^{5}+20 Z^{4}-24 Z^{3}+18 Z^{2}-7 Z +1, \mathit{index} =5\right)^{-n -1}}{97097}+\frac{43107 \mathit{RootOf} \left(3 Z^{6}-9 Z^{5}+20 Z^{4}-24 Z^{3}+18 Z^{2}-7 Z +1, \mathit{index} =6\right)^{-n -1}}{97097}-\frac{29737 \mathit{RootOf} \left(3 Z^{6}-9 Z^{5}+20 Z^{4}-24 Z^{3}+18 Z^{2}-7 Z +1, \mathit{index} =1\right)^{-n}}{14938}-\frac{29737 \mathit{RootOf} \left(3 Z^{6}-9 Z^{5}+20 Z^{4}-24 Z^{3}+18 Z^{2}-7 Z +1, \mathit{index} =2\right)^{-n}}{14938}-\frac{29737 \mathit{RootOf} \left(3 Z^{6}-9 Z^{5}+20 Z^{4}-24 Z^{3}+18 Z^{2}-7 Z +1, \mathit{index} =3\right)^{-n}}{14938}-\frac{29737 \mathit{RootOf} \left(3 Z^{6}-9 Z^{5}+20 Z^{4}-24 Z^{3}+18 Z^{2}-7 Z +1, \mathit{index} =4\right)^{-n}}{14938}-\frac{29737 \mathit{RootOf} \left(3 Z^{6}-9 Z^{5}+20 Z^{4}-24 Z^{3}+18 Z^{2}-7 Z +1, \mathit{index} =5\right)^{-n}}{14938}-\frac{29737 \mathit{RootOf} \left(3 Z^{6}-9 Z^{5}+20 Z^{4}-24 Z^{3}+18 Z^{2}-7 Z +1, \mathit{index} =6\right)^{-n}}{14938}+\left(\left\{\begin{array}{cc}\frac{1}{3} & n =0 \\ 0 & \text{otherwise} \end{array}\right.\right)\)
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This specification was found using the strategy pack "Point Placements" and has 76 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_{29}\! \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_{16}\! \left(x \right)+F_{23}\! \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_{21}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{19}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{18}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{25}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{7}\! \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_{22}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{26}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{31}\! \left(x \right)+F_{62}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{37}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)+F_{7}\! \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_{18}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{59}\! \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_{41}\! \left(x \right)+F_{44}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{4}\! \left(x \right) F_{41}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{45}\! \left(x \right)\\ F_{45}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{46}\! \left(x \right)+F_{53}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{4}\! \left(x \right) F_{47}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)+F_{51}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{49}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{4}\! \left(x \right) F_{48}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)+F_{55}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{4}\! \left(x \right) F_{54}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{52}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{4}\! \left(x \right) F_{57}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{52}\! \left(x \right)+F_{58}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{56}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{60}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{4}\! \left(x \right) F_{61}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{48}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{4}\! \left(x \right) F_{63}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{41}\! \left(x \right)+F_{64}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{65}\! \left(x \right)\\ F_{65}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{66}\! \left(x \right)+F_{70}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{4}\! \left(x \right) F_{67}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{48}\! \left(x \right)+F_{68}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{69}\! \left(x \right)+F_{72}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{70}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{4}\! \left(x \right) F_{71}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{69}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{73}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{4}\! \left(x \right) F_{74}\! \left(x \right)\\ F_{74}\! \left(x \right) &= F_{69}\! \left(x \right)+F_{75}\! \left(x \right)\\ F_{75}\! \left(x \right) &= F_{73}\! \left(x \right)\\ \end{align*}\)