Av(1342, 1423, 2143, 3214)
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Generating Function
\(\displaystyle \frac{\left(2 x -1\right) \left(x -1\right)^{3}}{2 x^{6}+x^{5}+6 x^{4}-14 x^{3}+13 x^{2}-6 x +1}\)
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Counting Sequence
1, 1, 2, 6, 20, 63, 187, 543, 1579, 4623, 13588, 39964, 117471, 345123, 1013802, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(2 x^{6}+x^{5}+6 x^{4}-14 x^{3}+13 x^{2}-6 x +1\right) F \! \left(x \right)-\left(2 x -1\right) \left(x -1\right)^{3} = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 63\)
\(\displaystyle a \! \left(n +6\right) = -2 a \! \left(n \right)-a \! \left(n +1\right)-6 a \! \left(n +2\right)+14 a \! \left(n +3\right)-13 a \! \left(n +4\right)+6 a \! \left(n +5\right), \quad n \geq 6\)
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Explicit Closed Form
\(\displaystyle -\frac{8917680 \mathit{RootOf} \left(2 Z^{6}+Z^{5}+6 Z^{4}-14 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =1\right)^{-n +4}}{42909577}-\frac{8917680 \mathit{RootOf} \left(2 Z^{6}+Z^{5}+6 Z^{4}-14 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =2\right)^{-n +4}}{42909577}-\frac{8917680 \mathit{RootOf} \left(2 Z^{6}+Z^{5}+6 Z^{4}-14 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =3\right)^{-n +4}}{42909577}-\frac{8917680 \mathit{RootOf} \left(2 Z^{6}+Z^{5}+6 Z^{4}-14 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =4\right)^{-n +4}}{42909577}-\frac{8917680 \mathit{RootOf} \left(2 Z^{6}+Z^{5}+6 Z^{4}-14 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =5\right)^{-n +4}}{42909577}-\frac{8917680 \mathit{RootOf} \left(2 Z^{6}+Z^{5}+6 Z^{4}-14 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =6\right)^{-n +4}}{42909577}-\frac{9716004 \mathit{RootOf} \left(2 Z^{6}+Z^{5}+6 Z^{4}-14 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =1\right)^{-n +3}}{42909577}-\frac{9716004 \mathit{RootOf} \left(2 Z^{6}+Z^{5}+6 Z^{4}-14 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =2\right)^{-n +3}}{42909577}-\frac{9716004 \mathit{RootOf} \left(2 Z^{6}+Z^{5}+6 Z^{4}-14 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =3\right)^{-n +3}}{42909577}-\frac{9716004 \mathit{RootOf} \left(2 Z^{6}+Z^{5}+6 Z^{4}-14 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =4\right)^{-n +3}}{42909577}-\frac{9716004 \mathit{RootOf} \left(2 Z^{6}+Z^{5}+6 Z^{4}-14 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =5\right)^{-n +3}}{42909577}-\frac{9716004 \mathit{RootOf} \left(2 Z^{6}+Z^{5}+6 Z^{4}-14 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =6\right)^{-n +3}}{42909577}-\frac{32646896 \mathit{RootOf} \left(2 Z^{6}+Z^{5}+6 Z^{4}-14 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =1\right)^{-n +2}}{42909577}-\frac{32646896 \mathit{RootOf} \left(2 Z^{6}+Z^{5}+6 Z^{4}-14 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =2\right)^{-n +2}}{42909577}-\frac{32646896 \mathit{RootOf} \left(2 Z^{6}+Z^{5}+6 Z^{4}-14 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =3\right)^{-n +2}}{42909577}-\frac{32646896 \mathit{RootOf} \left(2 Z^{6}+Z^{5}+6 Z^{4}-14 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =4\right)^{-n +2}}{42909577}-\frac{32646896 \mathit{RootOf} \left(2 Z^{6}+Z^{5}+6 Z^{4}-14 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =5\right)^{-n +2}}{42909577}-\frac{32646896 \mathit{RootOf} \left(2 Z^{6}+Z^{5}+6 Z^{4}-14 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =6\right)^{-n +2}}{42909577}+\frac{43045833 \mathit{RootOf} \left(2 Z^{6}+Z^{5}+6 Z^{4}-14 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =1\right)^{-n +1}}{42909577}+\frac{43045833 \mathit{RootOf} \left(2 Z^{6}+Z^{5}+6 Z^{4}-14 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =2\right)^{-n +1}}{42909577}+\frac{43045833 \mathit{RootOf} \left(2 Z^{6}+Z^{5}+6 Z^{4}-14 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =3\right)^{-n +1}}{42909577}+\frac{43045833 \mathit{RootOf} \left(2 Z^{6}+Z^{5}+6 Z^{4}-14 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =4\right)^{-n +1}}{42909577}+\frac{43045833 \mathit{RootOf} \left(2 Z^{6}+Z^{5}+6 Z^{4}-14 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =5\right)^{-n +1}}{42909577}+\frac{43045833 \mathit{RootOf} \left(2 Z^{6}+Z^{5}+6 Z^{4}-14 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =6\right)^{-n +1}}{42909577}+\frac{9464610 \mathit{RootOf} \left(2 Z^{6}+Z^{5}+6 Z^{4}-14 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =1\right)^{-n -1}}{42909577}+\frac{9464610 \mathit{RootOf} \left(2 Z^{6}+Z^{5}+6 Z^{4}-14 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =2\right)^{-n -1}}{42909577}+\frac{9464610 \mathit{RootOf} \left(2 Z^{6}+Z^{5}+6 Z^{4}-14 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =3\right)^{-n -1}}{42909577}+\frac{9464610 \mathit{RootOf} \left(2 Z^{6}+Z^{5}+6 Z^{4}-14 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =4\right)^{-n -1}}{42909577}+\frac{9464610 \mathit{RootOf} \left(2 Z^{6}+Z^{5}+6 Z^{4}-14 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =5\right)^{-n -1}}{42909577}+\frac{9464610 \mathit{RootOf} \left(2 Z^{6}+Z^{5}+6 Z^{4}-14 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =6\right)^{-n -1}}{42909577}-\frac{25985977 \mathit{RootOf} \left(2 Z^{6}+Z^{5}+6 Z^{4}-14 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =1\right)^{-n}}{42909577}-\frac{25985977 \mathit{RootOf} \left(2 Z^{6}+Z^{5}+6 Z^{4}-14 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =2\right)^{-n}}{42909577}-\frac{25985977 \mathit{RootOf} \left(2 Z^{6}+Z^{5}+6 Z^{4}-14 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =3\right)^{-n}}{42909577}-\frac{25985977 \mathit{RootOf} \left(2 Z^{6}+Z^{5}+6 Z^{4}-14 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =4\right)^{-n}}{42909577}-\frac{25985977 \mathit{RootOf} \left(2 Z^{6}+Z^{5}+6 Z^{4}-14 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =5\right)^{-n}}{42909577}-\frac{25985977 \mathit{RootOf} \left(2 Z^{6}+Z^{5}+6 Z^{4}-14 Z^{3}+13 Z^{2}-6 Z +1, \mathit{index} =6\right)^{-n}}{42909577}\)
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This specification was found using the strategy pack "Point Placements" and has 79 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_{20}\! \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_{17}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{17}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{23}\! \left(x \right)+F_{61}\! \left(x \right)\\ F_{22}\! \left(x \right) &= 0\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{27}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{25}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{30}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{31}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{36}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{35}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{33}\! \left(x \right)\\ F_{36}\! \left(x \right) &= 2 F_{22}\! \left(x \right)+F_{37}\! \left(x \right)+F_{39}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{38}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{32}\! \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_{58}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{42}\! \left(x \right)+F_{56}\! \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_{49}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{45}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{46}\! \left(x \right)+F_{48}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{4}\! \left(x \right) F_{47}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{44}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{11}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)+F_{52}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{2}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{52}\! \left(x \right) &= 2 F_{22}\! \left(x \right)+F_{53}\! \left(x \right)+F_{55}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{4}\! \left(x \right) F_{54}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{49}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{4}\! \left(x \right) F_{41}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{4}\! \left(x \right) F_{57}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{41}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{59}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{4}\! \left(x \right) F_{60}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{33}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{4}\! \left(x \right) F_{62}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{20}\! \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) &= 2 F_{22}\! \left(x \right)+F_{59}\! \left(x \right)+F_{65}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{4}\! \left(x \right) F_{66}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{67}\! \left(x \right)+F_{73}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{68}\! \left(x \right)+F_{70}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{69}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{17}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{71}\! \left(x \right)\\ F_{71}\! \left(x \right) &= F_{4}\! \left(x \right) F_{72}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{67}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{74}\! \left(x \right)+F_{76}\! \left(x \right)\\ F_{74}\! \left(x \right) &= F_{75}\! \left(x \right)\\ F_{75}\! \left(x \right) &= F_{33}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{77}\! \left(x \right)\\ F_{77}\! \left(x \right) &= F_{4}\! \left(x \right) F_{78}\! \left(x \right)\\ F_{78}\! \left(x \right) &= F_{73}\! \left(x \right)\\ \end{align*}\)