Av(1342, 2143, 3214)
View Raw Data
Generating Function
\(\displaystyle -\frac{\left(x^{2}-3 x +1\right) \left(x -1\right)^{2}}{x^{5}-3 x^{4}+11 x^{3}-12 x^{2}+6 x -1}\)
Counting Sequence
1, 1, 2, 6, 21, 74, 253, 845, 2791, 9188, 30246, 99639, 328422, 1082797, 3570197, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{5}-3 x^{4}+11 x^{3}-12 x^{2}+6 x -1\right) F \! \left(x \right)+\left(x^{2}-3 x +1\right) \left(x -1\right)^{2} = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(n +5\right) = a \! \left(n \right)-3 a \! \left(n +1\right)+11 a \! \left(n +2\right)-12 a \! \left(n +3\right)+6 a \! \left(n +4\right), \quad n \geq 5\)
Explicit Closed Form
\(\displaystyle -\frac{948 \mathit{RootOf} \left(Z^{5}-3 Z^{4}+11 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =1\right)^{-n +3}}{26987}-\frac{948 \mathit{RootOf} \left(Z^{5}-3 Z^{4}+11 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =2\right)^{-n +3}}{26987}-\frac{948 \mathit{RootOf} \left(Z^{5}-3 Z^{4}+11 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =3\right)^{-n +3}}{26987}-\frac{948 \mathit{RootOf} \left(Z^{5}-3 Z^{4}+11 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =4\right)^{-n +3}}{26987}-\frac{948 \mathit{RootOf} \left(Z^{5}-3 Z^{4}+11 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =5\right)^{-n +3}}{26987}+\frac{2174 \mathit{RootOf} \left(Z^{5}-3 Z^{4}+11 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =1\right)^{-n +2}}{26987}+\frac{2174 \mathit{RootOf} \left(Z^{5}-3 Z^{4}+11 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =2\right)^{-n +2}}{26987}+\frac{2174 \mathit{RootOf} \left(Z^{5}-3 Z^{4}+11 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =3\right)^{-n +2}}{26987}+\frac{2174 \mathit{RootOf} \left(Z^{5}-3 Z^{4}+11 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =4\right)^{-n +2}}{26987}+\frac{2174 \mathit{RootOf} \left(Z^{5}-3 Z^{4}+11 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =5\right)^{-n +2}}{26987}-\frac{9347 \mathit{RootOf} \left(Z^{5}-3 Z^{4}+11 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =1\right)^{-n +1}}{26987}-\frac{9347 \mathit{RootOf} \left(Z^{5}-3 Z^{4}+11 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =2\right)^{-n +1}}{26987}-\frac{9347 \mathit{RootOf} \left(Z^{5}-3 Z^{4}+11 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =3\right)^{-n +1}}{26987}-\frac{9347 \mathit{RootOf} \left(Z^{5}-3 Z^{4}+11 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =4\right)^{-n +1}}{26987}-\frac{9347 \mathit{RootOf} \left(Z^{5}-3 Z^{4}+11 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =5\right)^{-n +1}}{26987}-\frac{858 \mathit{RootOf} \left(Z^{5}-3 Z^{4}+11 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =1\right)^{-n -1}}{26987}-\frac{858 \mathit{RootOf} \left(Z^{5}-3 Z^{4}+11 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =2\right)^{-n -1}}{26987}-\frac{858 \mathit{RootOf} \left(Z^{5}-3 Z^{4}+11 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =3\right)^{-n -1}}{26987}-\frac{858 \mathit{RootOf} \left(Z^{5}-3 Z^{4}+11 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =4\right)^{-n -1}}{26987}-\frac{858 \mathit{RootOf} \left(Z^{5}-3 Z^{4}+11 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =5\right)^{-n -1}}{26987}+\frac{10862 \mathit{RootOf} \left(Z^{5}-3 Z^{4}+11 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =1\right)^{-n}}{26987}+\frac{10862 \mathit{RootOf} \left(Z^{5}-3 Z^{4}+11 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =2\right)^{-n}}{26987}+\frac{10862 \mathit{RootOf} \left(Z^{5}-3 Z^{4}+11 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =3\right)^{-n}}{26987}+\frac{10862 \mathit{RootOf} \left(Z^{5}-3 Z^{4}+11 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =4\right)^{-n}}{26987}+\frac{10862 \mathit{RootOf} \left(Z^{5}-3 Z^{4}+11 Z^{3}-12 Z^{2}+6 Z -1, \mathit{index} =5\right)^{-n}}{26987}\)

This specification was found using the strategy pack "Point Placements" and has 63 rules.

Found on January 18, 2022.

Finding the specification took 2 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_{6}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{7}\! \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_{14}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{18}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{15}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{12}\! \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_{43}\! \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_{30}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{25}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{26}\! \left(x \right)+F_{28}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{27}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{24}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{34}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{33}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{31}\! \left(x \right)\\ F_{34}\! \left(x \right) &= 2 F_{21}\! \left(x \right)+F_{35}\! \left(x \right)+F_{37}\! \left(x \right)\\ F_{35}\! \left(x \right) &= F_{36}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{36}\! \left(x \right) &= F_{30}\! \left(x \right)\\ F_{37}\! \left(x \right) &= F_{38}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{39}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{4}\! \left(x \right) F_{41}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{39}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{4}\! \left(x \right) F_{44}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{45}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{46}\! \left(x \right)\\ F_{46}\! \left(x \right) &= 2 F_{21}\! \left(x \right)+F_{40}\! \left(x \right)+F_{47}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{4}\! \left(x \right) F_{48}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)+F_{56}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)+F_{53}\! \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_{15}\! \left(x \right)+F_{50}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{54}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{4}\! \left(x \right) F_{55}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{49}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)+F_{60}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{4}\! \left(x \right) F_{59}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{31}\! \left(x \right)+F_{57}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{61}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{4}\! \left(x \right) F_{62}\! \left(x \right)\\ F_{62}\! \left(x \right) &= F_{56}\! \left(x \right)\\ \end{align*}\)