Av(1243, 1342, 2143, 2413, 3124)
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Generating Function
\(\displaystyle \frac{\left(2 x -1\right) \left(x -1\right)^{4}}{x^{8}-x^{7}+8 x^{5}-21 x^{4}+27 x^{3}-19 x^{2}+7 x -1}\)
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Counting Sequence
1, 1, 2, 6, 19, 58, 173, 511, 1505, 4430, 13039, 38378, 112959, 332479, 978621, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(x^{8}-x^{7}+8 x^{5}-21 x^{4}+27 x^{3}-19 x^{2}+7 x -1\right) F \! \left(x \right)-\left(2 x -1\right) \left(x -1\right)^{4} = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 58\)
\(\displaystyle a \! \left(6\right) = 173\)
\(\displaystyle a \! \left(7\right) = 511\)
\(\displaystyle a \! \left(n +1\right) = a \! \left(n \right)+8 a \! \left(n +3\right)-21 a \! \left(n +4\right)+27 a \! \left(n +5\right)-19 a \! \left(n +6\right)+7 a \! \left(n +7\right)-a \! \left(n +8\right), \quad n \geq 8\)
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Explicit Closed Form
\(\displaystyle \frac{30105136 \mathit{RootOf} \left(Z^{8}-Z^{7}+8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =1\right)^{-n +6}}{240132811}+\frac{30105136 \mathit{RootOf} \left(Z^{8}-Z^{7}+8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =2\right)^{-n +6}}{240132811}+\frac{30105136 \mathit{RootOf} \left(Z^{8}-Z^{7}+8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =3\right)^{-n +6}}{240132811}+\frac{30105136 \mathit{RootOf} \left(Z^{8}-Z^{7}+8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =4\right)^{-n +6}}{240132811}+\frac{30105136 \mathit{RootOf} \left(Z^{8}-Z^{7}+8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =5\right)^{-n +6}}{240132811}+\frac{30105136 \mathit{RootOf} \left(Z^{8}-Z^{7}+8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =6\right)^{-n +6}}{240132811}+\frac{30105136 \mathit{RootOf} \left(Z^{8}-Z^{7}+8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =7\right)^{-n +6}}{240132811}+\frac{30105136 \mathit{RootOf} \left(Z^{8}-Z^{7}+8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =8\right)^{-n +6}}{240132811}-\frac{29856681 \mathit{RootOf} \left(Z^{8}-Z^{7}+8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =1\right)^{-n +5}}{240132811}-\frac{29856681 \mathit{RootOf} \left(Z^{8}-Z^{7}+8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =2\right)^{-n +5}}{240132811}-\frac{29856681 \mathit{RootOf} \left(Z^{8}-Z^{7}+8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =3\right)^{-n +5}}{240132811}-\frac{29856681 \mathit{RootOf} \left(Z^{8}-Z^{7}+8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =4\right)^{-n +5}}{240132811}-\frac{29856681 \mathit{RootOf} \left(Z^{8}-Z^{7}+8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =5\right)^{-n +5}}{240132811}-\frac{29856681 \mathit{RootOf} \left(Z^{8}-Z^{7}+8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =6\right)^{-n +5}}{240132811}-\frac{29856681 \mathit{RootOf} \left(Z^{8}-Z^{7}+8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =7\right)^{-n +5}}{240132811}-\frac{29856681 \mathit{RootOf} \left(Z^{8}-Z^{7}+8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =8\right)^{-n +5}}{240132811}-\frac{15686711 \mathit{RootOf} \left(Z^{8}-Z^{7}+8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =1\right)^{-n +4}}{240132811}-\frac{15686711 \mathit{RootOf} \left(Z^{8}-Z^{7}+8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =2\right)^{-n +4}}{240132811}-\frac{15686711 \mathit{RootOf} \left(Z^{8}-Z^{7}+8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =3\right)^{-n +4}}{240132811}-\frac{15686711 \mathit{RootOf} \left(Z^{8}-Z^{7}+8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =4\right)^{-n +4}}{240132811}-\frac{15686711 \mathit{RootOf} \left(Z^{8}-Z^{7}+8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =5\right)^{-n +4}}{240132811}-\frac{15686711 \mathit{RootOf} \left(Z^{8}-Z^{7}+8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =6\right)^{-n +4}}{240132811}-\frac{15686711 \mathit{RootOf} \left(Z^{8}-Z^{7}+8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =7\right)^{-n +4}}{240132811}-\frac{15686711 \mathit{RootOf} \left(Z^{8}-Z^{7}+8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =8\right)^{-n +4}}{240132811}+\frac{233756932 \mathit{RootOf} \left(Z^{8}-Z^{7}+8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =1\right)^{-n +3}}{240132811}+\frac{233756932 \mathit{RootOf} \left(Z^{8}-Z^{7}+8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =2\right)^{-n +3}}{240132811}+\frac{233756932 \mathit{RootOf} \left(Z^{8}-Z^{7}+8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =3\right)^{-n +3}}{240132811}+\frac{233756932 \mathit{RootOf} \left(Z^{8}-Z^{7}+8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =4\right)^{-n +3}}{240132811}+\frac{233756932 \mathit{RootOf} \left(Z^{8}-Z^{7}+8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =5\right)^{-n +3}}{240132811}+\frac{233756932 \mathit{RootOf} \left(Z^{8}-Z^{7}+8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =6\right)^{-n +3}}{240132811}+\frac{233756932 \mathit{RootOf} \left(Z^{8}-Z^{7}+8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =7\right)^{-n +3}}{240132811}+\frac{233756932 \mathit{RootOf} \left(Z^{8}-Z^{7}+8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =8\right)^{-n +3}}{240132811}-\frac{627883346 \mathit{RootOf} \left(Z^{8}-Z^{7}+8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =1\right)^{-n +2}}{240132811}-\frac{627883346 \mathit{RootOf} \left(Z^{8}-Z^{7}+8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =2\right)^{-n +2}}{240132811}-\frac{627883346 \mathit{RootOf} \left(Z^{8}-Z^{7}+8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =3\right)^{-n +2}}{240132811}-\frac{627883346 \mathit{RootOf} \left(Z^{8}-Z^{7}+8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =4\right)^{-n +2}}{240132811}-\frac{627883346 \mathit{RootOf} \left(Z^{8}-Z^{7}+8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =5\right)^{-n +2}}{240132811}-\frac{627883346 \mathit{RootOf} \left(Z^{8}-Z^{7}+8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =6\right)^{-n +2}}{240132811}-\frac{627883346 \mathit{RootOf} \left(Z^{8}-Z^{7}+8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =7\right)^{-n +2}}{240132811}-\frac{627883346 \mathit{RootOf} \left(Z^{8}-Z^{7}+8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =8\right)^{-n +2}}{240132811}+\frac{697618567 \mathit{RootOf} \left(Z^{8}-Z^{7}+8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =1\right)^{-n +1}}{240132811}+\frac{697618567 \mathit{RootOf} \left(Z^{8}-Z^{7}+8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =2\right)^{-n +1}}{240132811}+\frac{697618567 \mathit{RootOf} \left(Z^{8}-Z^{7}+8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =3\right)^{-n +1}}{240132811}+\frac{697618567 \mathit{RootOf} \left(Z^{8}-Z^{7}+8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =4\right)^{-n +1}}{240132811}+\frac{697618567 \mathit{RootOf} \left(Z^{8}-Z^{7}+8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =5\right)^{-n +1}}{240132811}+\frac{697618567 \mathit{RootOf} \left(Z^{8}-Z^{7}+8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =6\right)^{-n +1}}{240132811}+\frac{697618567 \mathit{RootOf} \left(Z^{8}-Z^{7}+8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =7\right)^{-n +1}}{240132811}+\frac{697618567 \mathit{RootOf} \left(Z^{8}-Z^{7}+8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =8\right)^{-n +1}}{240132811}+\frac{91919840 \mathit{RootOf} \left(Z^{8}-Z^{7}+8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =1\right)^{-n -1}}{240132811}+\frac{91919840 \mathit{RootOf} \left(Z^{8}-Z^{7}+8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =2\right)^{-n -1}}{240132811}+\frac{91919840 \mathit{RootOf} \left(Z^{8}-Z^{7}+8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =3\right)^{-n -1}}{240132811}+\frac{91919840 \mathit{RootOf} \left(Z^{8}-Z^{7}+8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =4\right)^{-n -1}}{240132811}+\frac{91919840 \mathit{RootOf} \left(Z^{8}-Z^{7}+8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =5\right)^{-n -1}}{240132811}+\frac{91919840 \mathit{RootOf} \left(Z^{8}-Z^{7}+8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =6\right)^{-n -1}}{240132811}+\frac{91919840 \mathit{RootOf} \left(Z^{8}-Z^{7}+8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =7\right)^{-n -1}}{240132811}+\frac{91919840 \mathit{RootOf} \left(Z^{8}-Z^{7}+8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =8\right)^{-n -1}}{240132811}-\frac{379849061 \mathit{RootOf} \left(Z^{8}-Z^{7}+8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =1\right)^{-n}}{240132811}-\frac{379849061 \mathit{RootOf} \left(Z^{8}-Z^{7}+8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =2\right)^{-n}}{240132811}-\frac{379849061 \mathit{RootOf} \left(Z^{8}-Z^{7}+8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =3\right)^{-n}}{240132811}-\frac{379849061 \mathit{RootOf} \left(Z^{8}-Z^{7}+8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =4\right)^{-n}}{240132811}-\frac{379849061 \mathit{RootOf} \left(Z^{8}-Z^{7}+8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =5\right)^{-n}}{240132811}-\frac{379849061 \mathit{RootOf} \left(Z^{8}-Z^{7}+8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =6\right)^{-n}}{240132811}-\frac{379849061 \mathit{RootOf} \left(Z^{8}-Z^{7}+8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =7\right)^{-n}}{240132811}-\frac{379849061 \mathit{RootOf} \left(Z^{8}-Z^{7}+8 Z^{5}-21 Z^{4}+27 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =8\right)^{-n}}{240132811}\)
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This specification was found using the strategy pack "Point Placements" and has 55 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_{19}\! \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_{16}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{18}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{11}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{14}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{20}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{22}\! \left(x \right)+F_{47}\! \left(x \right)\\ F_{21}\! \left(x \right) &= 0\\ F_{22}\! \left(x \right) &= F_{23}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{25}\! \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_{7}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{22}\! \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_{2}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{34}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{36}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{37}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)+F_{41}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{39}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{38}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{35}\! \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_{44}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)+F_{46}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{39}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{42}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{4}\! \left(x \right) F_{48}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{49}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{50}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{4}\! \left(x \right) F_{52}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{53}\! \left(x \right)+F_{54}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{29}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{50}\! \left(x \right)\\ \end{align*}\)