Av(1243, 1324, 2413, 3142, 3214)
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Generating Function
\(\displaystyle -\frac{\left(2 x -1\right) \left(x -1\right)^{3}}{x^{6}-x^{5}-7 x^{4}+14 x^{3}-13 x^{2}+6 x -1}\)
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Counting Sequence
1, 1, 2, 6, 19, 56, 159, 449, 1274, 3628, 10342, 29478, 84006, 239389, 682200, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(x^{6}-x^{5}-7 x^{4}+14 x^{3}-13 x^{2}+6 x -1\right) F \! \left(x \right)+\left(2 x -1\right) \left(x -1\right)^{3} = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 56\)
\(\displaystyle a \! \left(n +6\right) = a \! \left(n \right)-a \! \left(n +1\right)-7 a \! \left(n +2\right)+14 a \! \left(n +3\right)-13 a \! \left(n +4\right)+6 a \! \left(n +5\right), \quad n \geq 6\)
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Explicit Closed Form
\(\displaystyle \frac{122206 \mathit{RootOf} \left(Z^{6}-Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =1\right)^{-n +4}}{725803}+\frac{122206 \mathit{RootOf} \left(Z^{6}-Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =2\right)^{-n +4}}{725803}+\frac{122206 \mathit{RootOf} \left(Z^{6}-Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =3\right)^{-n +4}}{725803}+\frac{122206 \mathit{RootOf} \left(Z^{6}-Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =4\right)^{-n +4}}{725803}+\frac{122206 \mathit{RootOf} \left(Z^{6}-Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =5\right)^{-n +4}}{725803}+\frac{122206 \mathit{RootOf} \left(Z^{6}-Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =6\right)^{-n +4}}{725803}-\frac{46422 \mathit{RootOf} \left(Z^{6}-Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =1\right)^{-n +3}}{725803}-\frac{46422 \mathit{RootOf} \left(Z^{6}-Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =2\right)^{-n +3}}{725803}-\frac{46422 \mathit{RootOf} \left(Z^{6}-Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =3\right)^{-n +3}}{725803}-\frac{46422 \mathit{RootOf} \left(Z^{6}-Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =4\right)^{-n +3}}{725803}-\frac{46422 \mathit{RootOf} \left(Z^{6}-Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =5\right)^{-n +3}}{725803}-\frac{46422 \mathit{RootOf} \left(Z^{6}-Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =6\right)^{-n +3}}{725803}-\frac{896809 \mathit{RootOf} \left(Z^{6}-Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =1\right)^{-n +2}}{725803}-\frac{896809 \mathit{RootOf} \left(Z^{6}-Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =2\right)^{-n +2}}{725803}-\frac{896809 \mathit{RootOf} \left(Z^{6}-Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =3\right)^{-n +2}}{725803}-\frac{896809 \mathit{RootOf} \left(Z^{6}-Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =4\right)^{-n +2}}{725803}-\frac{896809 \mathit{RootOf} \left(Z^{6}-Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =5\right)^{-n +2}}{725803}-\frac{896809 \mathit{RootOf} \left(Z^{6}-Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =6\right)^{-n +2}}{725803}+\frac{1131010 \mathit{RootOf} \left(Z^{6}-Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =1\right)^{-n +1}}{725803}+\frac{1131010 \mathit{RootOf} \left(Z^{6}-Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =2\right)^{-n +1}}{725803}+\frac{1131010 \mathit{RootOf} \left(Z^{6}-Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =3\right)^{-n +1}}{725803}+\frac{1131010 \mathit{RootOf} \left(Z^{6}-Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =4\right)^{-n +1}}{725803}+\frac{1131010 \mathit{RootOf} \left(Z^{6}-Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =5\right)^{-n +1}}{725803}+\frac{1131010 \mathit{RootOf} \left(Z^{6}-Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =6\right)^{-n +1}}{725803}+\frac{222463 \mathit{RootOf} \left(Z^{6}-Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =1\right)^{-n -1}}{725803}+\frac{222463 \mathit{RootOf} \left(Z^{6}-Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =2\right)^{-n -1}}{725803}+\frac{222463 \mathit{RootOf} \left(Z^{6}-Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =3\right)^{-n -1}}{725803}+\frac{222463 \mathit{RootOf} \left(Z^{6}-Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =4\right)^{-n -1}}{725803}+\frac{222463 \mathit{RootOf} \left(Z^{6}-Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =5\right)^{-n -1}}{725803}+\frac{222463 \mathit{RootOf} \left(Z^{6}-Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =6\right)^{-n -1}}{725803}-\frac{707938 \mathit{RootOf} \left(Z^{6}-Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =1\right)^{-n}}{725803}-\frac{707938 \mathit{RootOf} \left(Z^{6}-Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =2\right)^{-n}}{725803}-\frac{707938 \mathit{RootOf} \left(Z^{6}-Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =3\right)^{-n}}{725803}-\frac{707938 \mathit{RootOf} \left(Z^{6}-Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =4\right)^{-n}}{725803}-\frac{707938 \mathit{RootOf} \left(Z^{6}-Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =5\right)^{-n}}{725803}-\frac{707938 \mathit{RootOf} \left(Z^{6}-Z^{5}-7 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =6\right)^{-n}}{725803}\)
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This specification was found using the strategy pack "Point Placements" and has 58 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_{16}\! \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_{15}\! \left(x \right) &= F_{13}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{44}\! \left(x \right)\\ F_{18}\! \left(x \right) &= 0\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{7}\! \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_{25}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{26}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{25}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{38}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)+F_{35}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{34}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{32}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{36}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{35}\! \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_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{33}\! \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_{4}\! \left(x \right) F_{45}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{55}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{47}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{48}\! \left(x \right)+F_{54}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{4}\! \left(x \right) F_{49}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)+F_{52}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{51}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{23}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{53}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{39}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{4}\! \left(x \right) F_{46}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{56}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{4}\! \left(x \right) F_{55}\! \left(x \right)\\ \end{align*}\)