Av(1243, 1432, 2314, 3214)
Generating Function
\(\displaystyle -\frac{\left(2 x -1\right) \left(x -1\right)^{3}}{2 x^{7}-x^{6}-2 x^{5}-6 x^{4}+14 x^{3}-13 x^{2}+6 x -1}\)
Counting Sequence
1, 1, 2, 6, 20, 62, 181, 521, 1509, 4401, 12865, 37593, 109774, 320462, 935553, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(2 x^{7}-x^{6}-2 x^{5}-6 x^{4}+14 x^{3}-13 x^{2}+6 x -1\right) F \! \left(x \right)+\left(2 x -1\right) \left(x -1\right)^{3} = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 62\)
\(\displaystyle a \! \left(6\right) = 181\)
\(\displaystyle a \! \left(n +7\right) = 2 a \! \left(n \right)-a \! \left(n +1\right)-2 a \! \left(n +2\right)-6 a \! \left(n +3\right)+14 a \! \left(n +4\right)-13 a \! \left(n +5\right)+6 a \! \left(n +6\right), \quad n \geq 7\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 62\)
\(\displaystyle a \! \left(6\right) = 181\)
\(\displaystyle a \! \left(n +7\right) = 2 a \! \left(n \right)-a \! \left(n +1\right)-2 a \! \left(n +2\right)-6 a \! \left(n +3\right)+14 a \! \left(n +4\right)-13 a \! \left(n +5\right)+6 a \! \left(n +6\right), \quad n \geq 7\)
Explicit Closed Form
\(\displaystyle \frac{204922 \mathit{RootOf} \left(2 Z^{7}-Z^{6}-2 Z^{5}-6 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =1\right)^{-n +5}}{798901}+\frac{204922 \mathit{RootOf} \left(2 Z^{7}-Z^{6}-2 Z^{5}-6 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =2\right)^{-n +5}}{798901}+\frac{204922 \mathit{RootOf} \left(2 Z^{7}-Z^{6}-2 Z^{5}-6 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =3\right)^{-n +5}}{798901}+\frac{204922 \mathit{RootOf} \left(2 Z^{7}-Z^{6}-2 Z^{5}-6 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =4\right)^{-n +5}}{798901}+\frac{204922 \mathit{RootOf} \left(2 Z^{7}-Z^{6}-2 Z^{5}-6 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =5\right)^{-n +5}}{798901}+\frac{204922 \mathit{RootOf} \left(2 Z^{7}-Z^{6}-2 Z^{5}-6 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =6\right)^{-n +5}}{798901}+\frac{204922 \mathit{RootOf} \left(2 Z^{7}-Z^{6}-2 Z^{5}-6 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =7\right)^{-n +5}}{798901}+\frac{9722 \mathit{RootOf} \left(2 Z^{7}-Z^{6}-2 Z^{5}-6 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =1\right)^{-n +4}}{798901}+\frac{9722 \mathit{RootOf} \left(2 Z^{7}-Z^{6}-2 Z^{5}-6 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =2\right)^{-n +4}}{798901}+\frac{9722 \mathit{RootOf} \left(2 Z^{7}-Z^{6}-2 Z^{5}-6 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =3\right)^{-n +4}}{798901}+\frac{9722 \mathit{RootOf} \left(2 Z^{7}-Z^{6}-2 Z^{5}-6 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =4\right)^{-n +4}}{798901}+\frac{9722 \mathit{RootOf} \left(2 Z^{7}-Z^{6}-2 Z^{5}-6 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =5\right)^{-n +4}}{798901}+\frac{9722 \mathit{RootOf} \left(2 Z^{7}-Z^{6}-2 Z^{5}-6 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =6\right)^{-n +4}}{798901}+\frac{9722 \mathit{RootOf} \left(2 Z^{7}-Z^{6}-2 Z^{5}-6 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =7\right)^{-n +4}}{798901}-\frac{435059 \mathit{RootOf} \left(2 Z^{7}-Z^{6}-2 Z^{5}-6 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =1\right)^{-n +3}}{1597802}-\frac{435059 \mathit{RootOf} \left(2 Z^{7}-Z^{6}-2 Z^{5}-6 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =2\right)^{-n +3}}{1597802}-\frac{435059 \mathit{RootOf} \left(2 Z^{7}-Z^{6}-2 Z^{5}-6 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =3\right)^{-n +3}}{1597802}-\frac{435059 \mathit{RootOf} \left(2 Z^{7}-Z^{6}-2 Z^{5}-6 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =4\right)^{-n +3}}{1597802}-\frac{435059 \mathit{RootOf} \left(2 Z^{7}-Z^{6}-2 Z^{5}-6 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =5\right)^{-n +3}}{1597802}-\frac{435059 \mathit{RootOf} \left(2 Z^{7}-Z^{6}-2 Z^{5}-6 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =6\right)^{-n +3}}{1597802}-\frac{435059 \mathit{RootOf} \left(2 Z^{7}-Z^{6}-2 Z^{5}-6 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =7\right)^{-n +3}}{1597802}-\frac{757563 \mathit{RootOf} \left(2 Z^{7}-Z^{6}-2 Z^{5}-6 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =1\right)^{-n +2}}{798901}-\frac{757563 \mathit{RootOf} \left(2 Z^{7}-Z^{6}-2 Z^{5}-6 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =2\right)^{-n +2}}{798901}-\frac{757563 \mathit{RootOf} \left(2 Z^{7}-Z^{6}-2 Z^{5}-6 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =3\right)^{-n +2}}{798901}-\frac{757563 \mathit{RootOf} \left(2 Z^{7}-Z^{6}-2 Z^{5}-6 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =4\right)^{-n +2}}{798901}-\frac{757563 \mathit{RootOf} \left(2 Z^{7}-Z^{6}-2 Z^{5}-6 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =5\right)^{-n +2}}{798901}-\frac{757563 \mathit{RootOf} \left(2 Z^{7}-Z^{6}-2 Z^{5}-6 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =6\right)^{-n +2}}{798901}-\frac{757563 \mathit{RootOf} \left(2 Z^{7}-Z^{6}-2 Z^{5}-6 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =7\right)^{-n +2}}{798901}+\frac{1013736 \mathit{RootOf} \left(2 Z^{7}-Z^{6}-2 Z^{5}-6 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =1\right)^{-n +1}}{798901}+\frac{1013736 \mathit{RootOf} \left(2 Z^{7}-Z^{6}-2 Z^{5}-6 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =2\right)^{-n +1}}{798901}+\frac{1013736 \mathit{RootOf} \left(2 Z^{7}-Z^{6}-2 Z^{5}-6 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =3\right)^{-n +1}}{798901}+\frac{1013736 \mathit{RootOf} \left(2 Z^{7}-Z^{6}-2 Z^{5}-6 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =4\right)^{-n +1}}{798901}+\frac{1013736 \mathit{RootOf} \left(2 Z^{7}-Z^{6}-2 Z^{5}-6 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =5\right)^{-n +1}}{798901}+\frac{1013736 \mathit{RootOf} \left(2 Z^{7}-Z^{6}-2 Z^{5}-6 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =6\right)^{-n +1}}{798901}+\frac{1013736 \mathit{RootOf} \left(2 Z^{7}-Z^{6}-2 Z^{5}-6 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =7\right)^{-n +1}}{798901}+\frac{403803 \mathit{RootOf} \left(2 Z^{7}-Z^{6}-2 Z^{5}-6 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =1\right)^{-n -1}}{1597802}+\frac{403803 \mathit{RootOf} \left(2 Z^{7}-Z^{6}-2 Z^{5}-6 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =2\right)^{-n -1}}{1597802}+\frac{403803 \mathit{RootOf} \left(2 Z^{7}-Z^{6}-2 Z^{5}-6 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =3\right)^{-n -1}}{1597802}+\frac{403803 \mathit{RootOf} \left(2 Z^{7}-Z^{6}-2 Z^{5}-6 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =4\right)^{-n -1}}{1597802}+\frac{403803 \mathit{RootOf} \left(2 Z^{7}-Z^{6}-2 Z^{5}-6 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =5\right)^{-n -1}}{1597802}+\frac{403803 \mathit{RootOf} \left(2 Z^{7}-Z^{6}-2 Z^{5}-6 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =6\right)^{-n -1}}{1597802}+\frac{403803 \mathit{RootOf} \left(2 Z^{7}-Z^{6}-2 Z^{5}-6 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =7\right)^{-n -1}}{1597802}-\frac{1223543 \mathit{RootOf} \left(2 Z^{7}-Z^{6}-2 Z^{5}-6 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =1\right)^{-n}}{1597802}-\frac{1223543 \mathit{RootOf} \left(2 Z^{7}-Z^{6}-2 Z^{5}-6 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =2\right)^{-n}}{1597802}-\frac{1223543 \mathit{RootOf} \left(2 Z^{7}-Z^{6}-2 Z^{5}-6 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =3\right)^{-n}}{1597802}-\frac{1223543 \mathit{RootOf} \left(2 Z^{7}-Z^{6}-2 Z^{5}-6 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =4\right)^{-n}}{1597802}-\frac{1223543 \mathit{RootOf} \left(2 Z^{7}-Z^{6}-2 Z^{5}-6 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =5\right)^{-n}}{1597802}-\frac{1223543 \mathit{RootOf} \left(2 Z^{7}-Z^{6}-2 Z^{5}-6 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =6\right)^{-n}}{1597802}-\frac{1223543 \mathit{RootOf} \left(2 Z^{7}-Z^{6}-2 Z^{5}-6 Z^{4}+14 Z^{3}-13 Z^{2}+6 Z -1, \mathit{index} =7\right)^{-n}}{1597802}\)
This specification was found using the strategy pack "Point Placements" and has 87 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_{22}\! \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_{16}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{15}\! \left(x \right) &= 0\\
F_{16}\! \left(x \right) &= F_{17}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{17}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{18}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{16}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{20}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{20}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{21}\! \left(x \right)\\
F_{21}\! \left(x \right) &= F_{19}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{23}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{24}\! \left(x \right)+F_{83}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{25}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{28}\! \left(x \right)+F_{35}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{30}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{31}\! \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_{21}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= F_{32}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{20}\! \left(x \right)+F_{37}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{38}\! \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_{18}\! \left(x \right)+F_{41}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{39}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{57}\! \left(x \right)\\
F_{43}\! \left(x \right) &= F_{44}\! \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_{49}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{2}\! \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_{47}\! \left(x \right)+F_{50}\! \left(x \right)\\
F_{50}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{51}\! \left(x \right)+F_{54}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{4}\! \left(x \right) F_{52}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{51}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{4}\! \left(x \right) F_{55}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{54}\! \left(x \right)\\
F_{57}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{58}\! \left(x \right)+F_{75}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{4}\! \left(x \right) F_{59}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{52}\! \left(x \right)+F_{60}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{61}\! \left(x \right)+F_{71}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{62}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{4}\! \left(x \right) F_{63}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{61}\! \left(x \right)+F_{64}\! \left(x \right)\\
F_{64}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{48}\! \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_{69}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{68}\! \left(x \right)\\
F_{68}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{19}\! \left(x \right)+F_{28}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{47}\! \left(x \right)+F_{70}\! \left(x \right)\\
F_{70}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{58}\! \left(x \right)+F_{62}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{72}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{4}\! \left(x \right) F_{73}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{61}\! \left(x \right)+F_{74}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{72}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{4}\! \left(x \right) F_{76}\! \left(x \right)\\
F_{76}\! \left(x \right) &= F_{63}\! \left(x \right)+F_{77}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{78}\! \left(x \right)+F_{79}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{51}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{80}\! \left(x \right)\\
F_{80}\! \left(x \right) &= F_{4}\! \left(x \right) F_{81}\! \left(x \right)\\
F_{81}\! \left(x \right) &= F_{78}\! \left(x \right)+F_{82}\! \left(x \right)\\
F_{82}\! \left(x \right) &= F_{80}\! \left(x \right)\\
F_{83}\! \left(x \right) &= F_{4}\! \left(x \right) F_{84}\! \left(x \right)\\
F_{84}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{85}\! \left(x \right)\\
F_{85}\! \left(x \right) &= F_{64}\! \left(x \right)+F_{86}\! \left(x \right)\\
F_{86}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{51}\! \left(x \right)+F_{62}\! \left(x \right)\\
\end{align*}\)