Av(1243, 1342, 1432, 3214)
Generating Function
\(\displaystyle \frac{\left(x^{2}+x -1\right) \left(x -1\right)^{3}}{x^{7}+2 x^{6}+3 x^{5}-3 x^{4}-5 x^{3}+8 x^{2}-5 x +1}\)
Counting Sequence
1, 1, 2, 6, 20, 63, 186, 535, 1539, 4460, 12986, 37846, 110219, 320764, 933263, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{7}+2 x^{6}+3 x^{5}-3 x^{4}-5 x^{3}+8 x^{2}-5 x +1\right) F \! \left(x \right)-\left(x^{2}+x -1\right) \left(x -1\right)^{3} = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 63\)
\(\displaystyle a \! \left(6\right) = 186\)
\(\displaystyle a \! \left(n +7\right) = -a \! \left(n \right)-2 a \! \left(n +1\right)-3 a \! \left(n +2\right)+3 a \! \left(n +3\right)+5 a \! \left(n +4\right)-8 a \! \left(n +5\right)+5 a \! \left(n +6\right), \quad n \geq 7\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 63\)
\(\displaystyle a \! \left(6\right) = 186\)
\(\displaystyle a \! \left(n +7\right) = -a \! \left(n \right)-2 a \! \left(n +1\right)-3 a \! \left(n +2\right)+3 a \! \left(n +3\right)+5 a \! \left(n +4\right)-8 a \! \left(n +5\right)+5 a \! \left(n +6\right), \quad n \geq 7\)
Explicit Closed Form
\(\displaystyle -\frac{7248725 \mathit{RootOf} \left(Z^{7}+2 Z^{6}+3 Z^{5}-3 Z^{4}-5 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =1\right)^{-n +5}}{152685722}-\frac{7248725 \mathit{RootOf} \left(Z^{7}+2 Z^{6}+3 Z^{5}-3 Z^{4}-5 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)^{-n +5}}{152685722}-\frac{7248725 \mathit{RootOf} \left(Z^{7}+2 Z^{6}+3 Z^{5}-3 Z^{4}-5 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)^{-n +5}}{152685722}-\frac{7248725 \mathit{RootOf} \left(Z^{7}+2 Z^{6}+3 Z^{5}-3 Z^{4}-5 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)^{-n +5}}{152685722}-\frac{7248725 \mathit{RootOf} \left(Z^{7}+2 Z^{6}+3 Z^{5}-3 Z^{4}-5 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =5\right)^{-n +5}}{152685722}-\frac{7248725 \mathit{RootOf} \left(Z^{7}+2 Z^{6}+3 Z^{5}-3 Z^{4}-5 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =6\right)^{-n +5}}{152685722}-\frac{7248725 \mathit{RootOf} \left(Z^{7}+2 Z^{6}+3 Z^{5}-3 Z^{4}-5 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =7\right)^{-n +5}}{152685722}-\frac{20683503 \mathit{RootOf} \left(Z^{7}+2 Z^{6}+3 Z^{5}-3 Z^{4}-5 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =1\right)^{-n +4}}{152685722}-\frac{20683503 \mathit{RootOf} \left(Z^{7}+2 Z^{6}+3 Z^{5}-3 Z^{4}-5 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)^{-n +4}}{152685722}-\frac{20683503 \mathit{RootOf} \left(Z^{7}+2 Z^{6}+3 Z^{5}-3 Z^{4}-5 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)^{-n +4}}{152685722}-\frac{20683503 \mathit{RootOf} \left(Z^{7}+2 Z^{6}+3 Z^{5}-3 Z^{4}-5 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)^{-n +4}}{152685722}-\frac{20683503 \mathit{RootOf} \left(Z^{7}+2 Z^{6}+3 Z^{5}-3 Z^{4}-5 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =5\right)^{-n +4}}{152685722}-\frac{20683503 \mathit{RootOf} \left(Z^{7}+2 Z^{6}+3 Z^{5}-3 Z^{4}-5 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =6\right)^{-n +4}}{152685722}-\frac{20683503 \mathit{RootOf} \left(Z^{7}+2 Z^{6}+3 Z^{5}-3 Z^{4}-5 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =7\right)^{-n +4}}{152685722}-\frac{18625293 \mathit{RootOf} \left(Z^{7}+2 Z^{6}+3 Z^{5}-3 Z^{4}-5 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =1\right)^{-n +3}}{76342861}-\frac{18625293 \mathit{RootOf} \left(Z^{7}+2 Z^{6}+3 Z^{5}-3 Z^{4}-5 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)^{-n +3}}{76342861}-\frac{18625293 \mathit{RootOf} \left(Z^{7}+2 Z^{6}+3 Z^{5}-3 Z^{4}-5 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)^{-n +3}}{76342861}-\frac{18625293 \mathit{RootOf} \left(Z^{7}+2 Z^{6}+3 Z^{5}-3 Z^{4}-5 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)^{-n +3}}{76342861}-\frac{18625293 \mathit{RootOf} \left(Z^{7}+2 Z^{6}+3 Z^{5}-3 Z^{4}-5 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =5\right)^{-n +3}}{76342861}-\frac{18625293 \mathit{RootOf} \left(Z^{7}+2 Z^{6}+3 Z^{5}-3 Z^{4}-5 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =6\right)^{-n +3}}{76342861}-\frac{18625293 \mathit{RootOf} \left(Z^{7}+2 Z^{6}+3 Z^{5}-3 Z^{4}-5 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =7\right)^{-n +3}}{76342861}-\frac{8988891 \mathit{RootOf} \left(Z^{7}+2 Z^{6}+3 Z^{5}-3 Z^{4}-5 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =1\right)^{-n +2}}{152685722}-\frac{8988891 \mathit{RootOf} \left(Z^{7}+2 Z^{6}+3 Z^{5}-3 Z^{4}-5 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)^{-n +2}}{152685722}-\frac{8988891 \mathit{RootOf} \left(Z^{7}+2 Z^{6}+3 Z^{5}-3 Z^{4}-5 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)^{-n +2}}{152685722}-\frac{8988891 \mathit{RootOf} \left(Z^{7}+2 Z^{6}+3 Z^{5}-3 Z^{4}-5 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)^{-n +2}}{152685722}-\frac{8988891 \mathit{RootOf} \left(Z^{7}+2 Z^{6}+3 Z^{5}-3 Z^{4}-5 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =5\right)^{-n +2}}{152685722}-\frac{8988891 \mathit{RootOf} \left(Z^{7}+2 Z^{6}+3 Z^{5}-3 Z^{4}-5 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =6\right)^{-n +2}}{152685722}-\frac{8988891 \mathit{RootOf} \left(Z^{7}+2 Z^{6}+3 Z^{5}-3 Z^{4}-5 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =7\right)^{-n +2}}{152685722}+\frac{2437208 \mathit{RootOf} \left(Z^{7}+2 Z^{6}+3 Z^{5}-3 Z^{4}-5 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =1\right)^{-n +1}}{10906123}+\frac{2437208 \mathit{RootOf} \left(Z^{7}+2 Z^{6}+3 Z^{5}-3 Z^{4}-5 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)^{-n +1}}{10906123}+\frac{2437208 \mathit{RootOf} \left(Z^{7}+2 Z^{6}+3 Z^{5}-3 Z^{4}-5 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)^{-n +1}}{10906123}+\frac{2437208 \mathit{RootOf} \left(Z^{7}+2 Z^{6}+3 Z^{5}-3 Z^{4}-5 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)^{-n +1}}{10906123}+\frac{2437208 \mathit{RootOf} \left(Z^{7}+2 Z^{6}+3 Z^{5}-3 Z^{4}-5 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =5\right)^{-n +1}}{10906123}+\frac{2437208 \mathit{RootOf} \left(Z^{7}+2 Z^{6}+3 Z^{5}-3 Z^{4}-5 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =6\right)^{-n +1}}{10906123}+\frac{2437208 \mathit{RootOf} \left(Z^{7}+2 Z^{6}+3 Z^{5}-3 Z^{4}-5 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =7\right)^{-n +1}}{10906123}+\frac{18047609 \mathit{RootOf} \left(Z^{7}+2 Z^{6}+3 Z^{5}-3 Z^{4}-5 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =1\right)^{-n -1}}{152685722}+\frac{18047609 \mathit{RootOf} \left(Z^{7}+2 Z^{6}+3 Z^{5}-3 Z^{4}-5 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)^{-n -1}}{152685722}+\frac{18047609 \mathit{RootOf} \left(Z^{7}+2 Z^{6}+3 Z^{5}-3 Z^{4}-5 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)^{-n -1}}{152685722}+\frac{18047609 \mathit{RootOf} \left(Z^{7}+2 Z^{6}+3 Z^{5}-3 Z^{4}-5 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)^{-n -1}}{152685722}+\frac{18047609 \mathit{RootOf} \left(Z^{7}+2 Z^{6}+3 Z^{5}-3 Z^{4}-5 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =5\right)^{-n -1}}{152685722}+\frac{18047609 \mathit{RootOf} \left(Z^{7}+2 Z^{6}+3 Z^{5}-3 Z^{4}-5 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =6\right)^{-n -1}}{152685722}+\frac{18047609 \mathit{RootOf} \left(Z^{7}+2 Z^{6}+3 Z^{5}-3 Z^{4}-5 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =7\right)^{-n -1}}{152685722}-\frac{7855859 \mathit{RootOf} \left(Z^{7}+2 Z^{6}+3 Z^{5}-3 Z^{4}-5 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =1\right)^{-n}}{76342861}-\frac{7855859 \mathit{RootOf} \left(Z^{7}+2 Z^{6}+3 Z^{5}-3 Z^{4}-5 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =2\right)^{-n}}{76342861}-\frac{7855859 \mathit{RootOf} \left(Z^{7}+2 Z^{6}+3 Z^{5}-3 Z^{4}-5 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =3\right)^{-n}}{76342861}-\frac{7855859 \mathit{RootOf} \left(Z^{7}+2 Z^{6}+3 Z^{5}-3 Z^{4}-5 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =4\right)^{-n}}{76342861}-\frac{7855859 \mathit{RootOf} \left(Z^{7}+2 Z^{6}+3 Z^{5}-3 Z^{4}-5 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =5\right)^{-n}}{76342861}-\frac{7855859 \mathit{RootOf} \left(Z^{7}+2 Z^{6}+3 Z^{5}-3 Z^{4}-5 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =6\right)^{-n}}{76342861}-\frac{7855859 \mathit{RootOf} \left(Z^{7}+2 Z^{6}+3 Z^{5}-3 Z^{4}-5 Z^{3}+8 Z^{2}-5 Z +1, \mathit{index} =7\right)^{-n}}{76342861}\)
This specification was found using the strategy pack "Point Placements" and has 81 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_{17}\! \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_{11}\! \left(x \right)+F_{14}\! \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_{11}\! \left(x \right)+F_{14}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{2}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{20}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{19}\! \left(x \right) &= 0\\
F_{20}\! \left(x \right) &= F_{21}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{23}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{24}\! \left(x \right)+F_{26}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{13}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{29}\! \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_{14}\! \left(x \right)+F_{29}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)+F_{43}\! \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_{36}\! \left(x \right)+F_{39}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{38}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{36}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{39}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{40}\! \left(x \right) &= F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{4}\! \left(x \right) F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{40}\! \left(x \right)\\
F_{43}\! \left(x \right) &= 2 F_{19}\! \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_{46}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= F_{48}\! \left(x \right)\\
F_{48}\! \left(x \right) &= F_{4}\! \left(x \right) F_{46}\! \left(x \right)\\
F_{49}\! \left(x \right) &= F_{4}\! \left(x \right) F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{51}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{53}\! \left(x \right)+F_{61}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{4}\! \left(x \right) F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{55}\! \left(x \right)+F_{58}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{19}\! \left(x \right)+F_{24}\! \left(x \right)+F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{13}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= 2 F_{19}\! \left(x \right)+F_{44}\! \left(x \right)+F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{4}\! \left(x \right) F_{51}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{4}\! \left(x \right) F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{52}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{64}\! \left(x \right)\\
F_{64}\! \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_{40}\! \left(x \right)+F_{64}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{4}\! \left(x \right) F_{68}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{62}\! \left(x \right)+F_{69}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{52}\! \left(x \right)+F_{70}\! \left(x \right)\\
F_{70}\! \left(x \right) &= 2 F_{19}\! \left(x \right)+F_{48}\! \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_{73}\! \left(x \right)+F_{77}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{74}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{75}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{4}\! \left(x \right) F_{76}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{31}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{40}\! \left(x \right)+F_{78}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{79}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{4}\! \left(x \right) F_{80}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{66}\! \left(x \right)\\
\end{align*}\)