Av(1234, 2413, 3142)
Generating Function
\(\displaystyle -\frac{\left(x^{3}-2 x^{2}+3 x -1\right) \left(x -1\right)^{3}}{x^{7}-4 x^{6}+12 x^{5}-23 x^{4}+28 x^{3}-19 x^{2}+7 x -1}\)
Counting Sequence
1, 1, 2, 6, 21, 73, 243, 785, 2504, 7968, 25389, 81033, 258873, 827263, 2643616, ...
Implicit Equation for the Generating Function
\(\displaystyle \left(x^{7}-4 x^{6}+12 x^{5}-23 x^{4}+28 x^{3}-19 x^{2}+7 x -1\right) F \! \left(x \right)+\left(x^{3}-2 x^{2}+3 x -1\right) \left(x -1\right)^{3} = 0\)
Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 73\)
\(\displaystyle a \! \left(6\right) = 243\)
\(\displaystyle a \! \left(n +7\right) = a \! \left(n \right)-4 a \! \left(n +1\right)+12 a \! \left(n +2\right)-23 a \! \left(n +3\right)+28 a \! \left(n +4\right)-19 a \! \left(n +5\right)+7 a \! \left(n +6\right), \quad n \geq 7\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 21\)
\(\displaystyle a \! \left(5\right) = 73\)
\(\displaystyle a \! \left(6\right) = 243\)
\(\displaystyle a \! \left(n +7\right) = a \! \left(n \right)-4 a \! \left(n +1\right)+12 a \! \left(n +2\right)-23 a \! \left(n +3\right)+28 a \! \left(n +4\right)-19 a \! \left(n +5\right)+7 a \! \left(n +6\right), \quad n \geq 7\)
Explicit Closed Form
\(\displaystyle -\frac{12189 \mathit{RootOf} \left(Z^{7}-4 Z^{6}+12 Z^{5}-23 Z^{4}+28 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =1\right)^{-n +5}}{311981}-\frac{12189 \mathit{RootOf} \left(Z^{7}-4 Z^{6}+12 Z^{5}-23 Z^{4}+28 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =2\right)^{-n +5}}{311981}-\frac{12189 \mathit{RootOf} \left(Z^{7}-4 Z^{6}+12 Z^{5}-23 Z^{4}+28 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =3\right)^{-n +5}}{311981}-\frac{12189 \mathit{RootOf} \left(Z^{7}-4 Z^{6}+12 Z^{5}-23 Z^{4}+28 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =4\right)^{-n +5}}{311981}-\frac{12189 \mathit{RootOf} \left(Z^{7}-4 Z^{6}+12 Z^{5}-23 Z^{4}+28 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =5\right)^{-n +5}}{311981}-\frac{12189 \mathit{RootOf} \left(Z^{7}-4 Z^{6}+12 Z^{5}-23 Z^{4}+28 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =6\right)^{-n +5}}{311981}-\frac{12189 \mathit{RootOf} \left(Z^{7}-4 Z^{6}+12 Z^{5}-23 Z^{4}+28 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =7\right)^{-n +5}}{311981}+\frac{56185 \mathit{RootOf} \left(Z^{7}-4 Z^{6}+12 Z^{5}-23 Z^{4}+28 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =1\right)^{-n +4}}{311981}+\frac{56185 \mathit{RootOf} \left(Z^{7}-4 Z^{6}+12 Z^{5}-23 Z^{4}+28 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =2\right)^{-n +4}}{311981}+\frac{56185 \mathit{RootOf} \left(Z^{7}-4 Z^{6}+12 Z^{5}-23 Z^{4}+28 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =3\right)^{-n +4}}{311981}+\frac{56185 \mathit{RootOf} \left(Z^{7}-4 Z^{6}+12 Z^{5}-23 Z^{4}+28 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =4\right)^{-n +4}}{311981}+\frac{56185 \mathit{RootOf} \left(Z^{7}-4 Z^{6}+12 Z^{5}-23 Z^{4}+28 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =5\right)^{-n +4}}{311981}+\frac{56185 \mathit{RootOf} \left(Z^{7}-4 Z^{6}+12 Z^{5}-23 Z^{4}+28 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =6\right)^{-n +4}}{311981}+\frac{56185 \mathit{RootOf} \left(Z^{7}-4 Z^{6}+12 Z^{5}-23 Z^{4}+28 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =7\right)^{-n +4}}{311981}-\frac{304023 \mathit{RootOf} \left(Z^{7}-4 Z^{6}+12 Z^{5}-23 Z^{4}+28 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =1\right)^{-n +3}}{623962}-\frac{304023 \mathit{RootOf} \left(Z^{7}-4 Z^{6}+12 Z^{5}-23 Z^{4}+28 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =2\right)^{-n +3}}{623962}-\frac{304023 \mathit{RootOf} \left(Z^{7}-4 Z^{6}+12 Z^{5}-23 Z^{4}+28 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =3\right)^{-n +3}}{623962}-\frac{304023 \mathit{RootOf} \left(Z^{7}-4 Z^{6}+12 Z^{5}-23 Z^{4}+28 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =4\right)^{-n +3}}{623962}-\frac{304023 \mathit{RootOf} \left(Z^{7}-4 Z^{6}+12 Z^{5}-23 Z^{4}+28 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =5\right)^{-n +3}}{623962}-\frac{304023 \mathit{RootOf} \left(Z^{7}-4 Z^{6}+12 Z^{5}-23 Z^{4}+28 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =6\right)^{-n +3}}{623962}-\frac{304023 \mathit{RootOf} \left(Z^{7}-4 Z^{6}+12 Z^{5}-23 Z^{4}+28 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =7\right)^{-n +3}}{623962}+\frac{607623 \mathit{RootOf} \left(Z^{7}-4 Z^{6}+12 Z^{5}-23 Z^{4}+28 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =1\right)^{-n +2}}{623962}+\frac{607623 \mathit{RootOf} \left(Z^{7}-4 Z^{6}+12 Z^{5}-23 Z^{4}+28 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =2\right)^{-n +2}}{623962}+\frac{607623 \mathit{RootOf} \left(Z^{7}-4 Z^{6}+12 Z^{5}-23 Z^{4}+28 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =3\right)^{-n +2}}{623962}+\frac{607623 \mathit{RootOf} \left(Z^{7}-4 Z^{6}+12 Z^{5}-23 Z^{4}+28 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =4\right)^{-n +2}}{623962}+\frac{607623 \mathit{RootOf} \left(Z^{7}-4 Z^{6}+12 Z^{5}-23 Z^{4}+28 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =5\right)^{-n +2}}{623962}+\frac{607623 \mathit{RootOf} \left(Z^{7}-4 Z^{6}+12 Z^{5}-23 Z^{4}+28 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =6\right)^{-n +2}}{623962}+\frac{607623 \mathit{RootOf} \left(Z^{7}-4 Z^{6}+12 Z^{5}-23 Z^{4}+28 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =7\right)^{-n +2}}{623962}-\frac{684809 \mathit{RootOf} \left(Z^{7}-4 Z^{6}+12 Z^{5}-23 Z^{4}+28 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =1\right)^{-n +1}}{623962}-\frac{684809 \mathit{RootOf} \left(Z^{7}-4 Z^{6}+12 Z^{5}-23 Z^{4}+28 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =2\right)^{-n +1}}{623962}-\frac{684809 \mathit{RootOf} \left(Z^{7}-4 Z^{6}+12 Z^{5}-23 Z^{4}+28 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =3\right)^{-n +1}}{623962}-\frac{684809 \mathit{RootOf} \left(Z^{7}-4 Z^{6}+12 Z^{5}-23 Z^{4}+28 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =4\right)^{-n +1}}{623962}-\frac{684809 \mathit{RootOf} \left(Z^{7}-4 Z^{6}+12 Z^{5}-23 Z^{4}+28 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =5\right)^{-n +1}}{623962}-\frac{684809 \mathit{RootOf} \left(Z^{7}-4 Z^{6}+12 Z^{5}-23 Z^{4}+28 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =6\right)^{-n +1}}{623962}-\frac{684809 \mathit{RootOf} \left(Z^{7}-4 Z^{6}+12 Z^{5}-23 Z^{4}+28 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =7\right)^{-n +1}}{623962}-\frac{37819 \mathit{RootOf} \left(Z^{7}-4 Z^{6}+12 Z^{5}-23 Z^{4}+28 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =1\right)^{-n -1}}{623962}-\frac{37819 \mathit{RootOf} \left(Z^{7}-4 Z^{6}+12 Z^{5}-23 Z^{4}+28 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =2\right)^{-n -1}}{623962}-\frac{37819 \mathit{RootOf} \left(Z^{7}-4 Z^{6}+12 Z^{5}-23 Z^{4}+28 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =3\right)^{-n -1}}{623962}-\frac{37819 \mathit{RootOf} \left(Z^{7}-4 Z^{6}+12 Z^{5}-23 Z^{4}+28 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =4\right)^{-n -1}}{623962}-\frac{37819 \mathit{RootOf} \left(Z^{7}-4 Z^{6}+12 Z^{5}-23 Z^{4}+28 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =5\right)^{-n -1}}{623962}-\frac{37819 \mathit{RootOf} \left(Z^{7}-4 Z^{6}+12 Z^{5}-23 Z^{4}+28 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =6\right)^{-n -1}}{623962}-\frac{37819 \mathit{RootOf} \left(Z^{7}-4 Z^{6}+12 Z^{5}-23 Z^{4}+28 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =7\right)^{-n -1}}{623962}+\frac{213366 \mathit{RootOf} \left(Z^{7}-4 Z^{6}+12 Z^{5}-23 Z^{4}+28 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =1\right)^{-n}}{311981}+\frac{213366 \mathit{RootOf} \left(Z^{7}-4 Z^{6}+12 Z^{5}-23 Z^{4}+28 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =2\right)^{-n}}{311981}+\frac{213366 \mathit{RootOf} \left(Z^{7}-4 Z^{6}+12 Z^{5}-23 Z^{4}+28 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =3\right)^{-n}}{311981}+\frac{213366 \mathit{RootOf} \left(Z^{7}-4 Z^{6}+12 Z^{5}-23 Z^{4}+28 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =4\right)^{-n}}{311981}+\frac{213366 \mathit{RootOf} \left(Z^{7}-4 Z^{6}+12 Z^{5}-23 Z^{4}+28 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =5\right)^{-n}}{311981}+\frac{213366 \mathit{RootOf} \left(Z^{7}-4 Z^{6}+12 Z^{5}-23 Z^{4}+28 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =6\right)^{-n}}{311981}+\frac{213366 \mathit{RootOf} \left(Z^{7}-4 Z^{6}+12 Z^{5}-23 Z^{4}+28 Z^{3}-19 Z^{2}+7 Z -1, \mathit{index} =7\right)^{-n}}{311981}\)
This specification was found using the strategy pack "Point Placements" and has 80 rules.
Found on January 18, 2022.Finding the specification took 5 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_{23}\! \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_{16}\! \left(x \right)+F_{20}\! \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_{18}\! \left(x \right)+F_{19}\! \left(x \right)\\
F_{18}\! \left(x \right) &= F_{11}\! \left(x \right)\\
F_{19}\! \left(x \right) &= F_{14}\! \left(x \right)\\
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_{7}\! \left(x \right)\\
F_{22}\! \left(x \right) &= F_{20}\! \left(x \right)\\
F_{23}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{24}\! \left(x \right)\\
F_{24}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{25}\! \left(x \right)+F_{60}\! \left(x \right)\\
F_{25}\! \left(x \right) &= F_{26}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{26}\! \left(x \right) &= F_{27}\! \left(x \right)+F_{33}\! \left(x \right)\\
F_{27}\! \left(x \right) &= F_{28}\! \left(x \right)+F_{7}\! \left(x \right)\\
F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)\\
F_{29}\! \left(x \right) &= F_{30}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{30}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{31}\! \left(x \right)\\
F_{31}\! \left(x \right) &= F_{32}\! \left(x \right)\\
F_{32}\! \left(x \right) &= F_{29}\! \left(x \right)\\
F_{33}\! \left(x \right) &= F_{24}\! \left(x \right)+F_{34}\! \left(x \right)\\
F_{34}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{35}\! \left(x \right)+F_{39}\! \left(x \right)\\
F_{35}\! \left(x \right) &= F_{36}\! \left(x \right) F_{4}\! \left(x \right)\\
F_{36}\! \left(x \right) &= F_{37}\! \left(x \right)+F_{38}\! \left(x \right)\\
F_{37}\! \left(x \right) &= F_{28}\! \left(x \right)\\
F_{38}\! \left(x \right) &= F_{34}\! \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_{55}\! \left(x \right)\\
F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)\\
F_{42}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{43}\! \left(x \right)+F_{52}\! \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_{11}\! \left(x \right)+F_{32}\! \left(x \right)\\
F_{46}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{47}\! \left(x \right)\\
F_{47}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{39}\! \left(x \right)+F_{48}\! \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_{51}\! \left(x \right)\\
F_{50}\! \left(x \right) &= F_{32}\! \left(x \right)\\
F_{51}\! \left(x \right) &= F_{47}\! \left(x \right)\\
F_{52}\! \left(x \right) &= F_{4}\! \left(x \right) F_{53}\! \left(x \right)\\
F_{53}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{54}\! \left(x \right)\\
F_{54}\! \left(x \right) &= F_{52}\! \left(x \right)\\
F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)\\
F_{56}\! \left(x \right) &= F_{4}\! \left(x \right) F_{57}\! \left(x \right)\\
F_{57}\! \left(x \right) &= F_{58}\! \left(x \right)+F_{59}\! \left(x \right)\\
F_{58}\! \left(x \right) &= F_{42}\! \left(x \right)\\
F_{59}\! \left(x \right) &= F_{56}\! \left(x \right)\\
F_{60}\! \left(x \right) &= F_{4}\! \left(x \right) F_{61}\! \left(x \right)\\
F_{61}\! \left(x \right) &= F_{62}\! \left(x \right)+F_{63}\! \left(x \right)\\
F_{62}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{42}\! \left(x \right)\\
F_{63}\! \left(x \right) &= F_{64}\! \left(x \right)+F_{76}\! \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_{53}\! \left(x \right)+F_{67}\! \left(x \right)\\
F_{67}\! \left(x \right) &= F_{64}\! \left(x \right)+F_{68}\! \left(x \right)\\
F_{68}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{69}\! \left(x \right)+F_{73}\! \left(x \right)\\
F_{69}\! \left(x \right) &= F_{4}\! \left(x \right) F_{70}\! \left(x \right)\\
F_{70}\! \left(x \right) &= F_{71}\! \left(x \right)+F_{72}\! \left(x \right)\\
F_{71}\! \left(x \right) &= F_{54}\! \left(x \right)\\
F_{72}\! \left(x \right) &= F_{68}\! \left(x \right)\\
F_{73}\! \left(x \right) &= F_{4}\! \left(x \right) F_{74}\! \left(x \right)\\
F_{74}\! \left(x \right) &= F_{64}\! \left(x \right)+F_{75}\! \left(x \right)\\
F_{75}\! \left(x \right) &= F_{73}\! \left(x \right)\\
F_{76}\! \left(x \right) &= 2 F_{15}\! \left(x \right)+F_{73}\! \left(x \right)+F_{77}\! \left(x \right)\\
F_{77}\! \left(x \right) &= F_{4}\! \left(x \right) F_{78}\! \left(x \right)\\
F_{78}\! \left(x \right) &= F_{58}\! \left(x \right)+F_{79}\! \left(x \right)\\
F_{79}\! \left(x \right) &= F_{76}\! \left(x \right)\\
\end{align*}\)