Av(1234, 1342, 1432, 3214)
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Generating Function
\(\displaystyle -\frac{2 x^{3}+x^{2}-1}{10 x^{9}+22 x^{8}+13 x^{7}-6 x^{6}-14 x^{5}-6 x^{4}-4 x^{3}-2 x^{2}-x +1}\)
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Counting Sequence
1, 1, 2, 6, 20, 60, 156, 413, 1146, 3214, 8922, 24544, 67558, 186592, 515887, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(10 x^{9}+22 x^{8}+13 x^{7}-6 x^{6}-14 x^{5}-6 x^{4}-4 x^{3}-2 x^{2}-x +1\right) F \! \left(x \right)+2 x^{3}+x^{2}-1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 20\)
\(\displaystyle a \! \left(5\right) = 60\)
\(\displaystyle a \! \left(6\right) = 156\)
\(\displaystyle a \! \left(7\right) = 413\)
\(\displaystyle a \! \left(8\right) = 1146\)
\(\displaystyle a \! \left(n +9\right) = -10 a \! \left(n \right)-22 a \! \left(n +1\right)-13 a \! \left(n +2\right)+6 a \! \left(n +3\right)+14 a \! \left(n +4\right)+6 a \! \left(n +5\right)+4 a \! \left(n +6\right)+2 a \! \left(n +7\right)+a \! \left(n +8\right), \quad n \geq 9\)
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Explicit Closed Form
\(\displaystyle \frac{14067721383280 \mathit{RootOf} \left(10 Z^{9}+22 Z^{8}+13 Z^{7}-6 Z^{6}-14 Z^{5}-6 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =1\right)^{-n +7}}{3641177375435009}+\frac{14067721383280 \mathit{RootOf} \left(10 Z^{9}+22 Z^{8}+13 Z^{7}-6 Z^{6}-14 Z^{5}-6 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =2\right)^{-n +7}}{3641177375435009}+\frac{14067721383280 \mathit{RootOf} \left(10 Z^{9}+22 Z^{8}+13 Z^{7}-6 Z^{6}-14 Z^{5}-6 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =3\right)^{-n +7}}{3641177375435009}+\frac{14067721383280 \mathit{RootOf} \left(10 Z^{9}+22 Z^{8}+13 Z^{7}-6 Z^{6}-14 Z^{5}-6 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =4\right)^{-n +7}}{3641177375435009}+\frac{14067721383280 \mathit{RootOf} \left(10 Z^{9}+22 Z^{8}+13 Z^{7}-6 Z^{6}-14 Z^{5}-6 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =5\right)^{-n +7}}{3641177375435009}+\frac{14067721383280 \mathit{RootOf} \left(10 Z^{9}+22 Z^{8}+13 Z^{7}-6 Z^{6}-14 Z^{5}-6 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =6\right)^{-n +7}}{3641177375435009}+\frac{14067721383280 \mathit{RootOf} \left(10 Z^{9}+22 Z^{8}+13 Z^{7}-6 Z^{6}-14 Z^{5}-6 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =7\right)^{-n +7}}{3641177375435009}+\frac{14067721383280 \mathit{RootOf} \left(10 Z^{9}+22 Z^{8}+13 Z^{7}-6 Z^{6}-14 Z^{5}-6 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =8\right)^{-n +7}}{3641177375435009}+\frac{14067721383280 \mathit{RootOf} \left(10 Z^{9}+22 Z^{8}+13 Z^{7}-6 Z^{6}-14 Z^{5}-6 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =9\right)^{-n +7}}{3641177375435009}+\frac{885670624154376 \mathit{RootOf} \left(10 Z^{9}+22 Z^{8}+13 Z^{7}-6 Z^{6}-14 Z^{5}-6 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =1\right)^{-n +6}}{3641177375435009}+\frac{885670624154376 \mathit{RootOf} \left(10 Z^{9}+22 Z^{8}+13 Z^{7}-6 Z^{6}-14 Z^{5}-6 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =2\right)^{-n +6}}{3641177375435009}+\frac{885670624154376 \mathit{RootOf} \left(10 Z^{9}+22 Z^{8}+13 Z^{7}-6 Z^{6}-14 Z^{5}-6 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =3\right)^{-n +6}}{3641177375435009}+\frac{885670624154376 \mathit{RootOf} \left(10 Z^{9}+22 Z^{8}+13 Z^{7}-6 Z^{6}-14 Z^{5}-6 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =4\right)^{-n +6}}{3641177375435009}+\frac{885670624154376 \mathit{RootOf} \left(10 Z^{9}+22 Z^{8}+13 Z^{7}-6 Z^{6}-14 Z^{5}-6 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =5\right)^{-n +6}}{3641177375435009}+\frac{885670624154376 \mathit{RootOf} \left(10 Z^{9}+22 Z^{8}+13 Z^{7}-6 Z^{6}-14 Z^{5}-6 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =6\right)^{-n +6}}{3641177375435009}+\frac{885670624154376 \mathit{RootOf} \left(10 Z^{9}+22 Z^{8}+13 Z^{7}-6 Z^{6}-14 Z^{5}-6 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =7\right)^{-n +6}}{3641177375435009}+\frac{885670624154376 \mathit{RootOf} \left(10 Z^{9}+22 Z^{8}+13 Z^{7}-6 Z^{6}-14 Z^{5}-6 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =8\right)^{-n +6}}{3641177375435009}+\frac{885670624154376 \mathit{RootOf} \left(10 Z^{9}+22 Z^{8}+13 Z^{7}-6 Z^{6}-14 Z^{5}-6 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =9\right)^{-n +6}}{3641177375435009}+\frac{373496119651216 \mathit{RootOf} \left(10 Z^{9}+22 Z^{8}+13 Z^{7}-6 Z^{6}-14 Z^{5}-6 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =1\right)^{-n +5}}{3641177375435009}+\frac{373496119651216 \mathit{RootOf} \left(10 Z^{9}+22 Z^{8}+13 Z^{7}-6 Z^{6}-14 Z^{5}-6 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =2\right)^{-n +5}}{3641177375435009}+\frac{373496119651216 \mathit{RootOf} \left(10 Z^{9}+22 Z^{8}+13 Z^{7}-6 Z^{6}-14 Z^{5}-6 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =3\right)^{-n +5}}{3641177375435009}+\frac{373496119651216 \mathit{RootOf} \left(10 Z^{9}+22 Z^{8}+13 Z^{7}-6 Z^{6}-14 Z^{5}-6 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =4\right)^{-n +5}}{3641177375435009}+\frac{373496119651216 \mathit{RootOf} \left(10 Z^{9}+22 Z^{8}+13 Z^{7}-6 Z^{6}-14 Z^{5}-6 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =5\right)^{-n +5}}{3641177375435009}+\frac{373496119651216 \mathit{RootOf} \left(10 Z^{9}+22 Z^{8}+13 Z^{7}-6 Z^{6}-14 Z^{5}-6 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =6\right)^{-n +5}}{3641177375435009}+\frac{373496119651216 \mathit{RootOf} \left(10 Z^{9}+22 Z^{8}+13 Z^{7}-6 Z^{6}-14 Z^{5}-6 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =7\right)^{-n +5}}{3641177375435009}+\frac{373496119651216 \mathit{RootOf} \left(10 Z^{9}+22 Z^{8}+13 Z^{7}-6 Z^{6}-14 Z^{5}-6 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =8\right)^{-n +5}}{3641177375435009}+\frac{373496119651216 \mathit{RootOf} \left(10 Z^{9}+22 Z^{8}+13 Z^{7}-6 Z^{6}-14 Z^{5}-6 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =9\right)^{-n +5}}{3641177375435009}-\frac{1463038124051290 \mathit{RootOf} \left(10 Z^{9}+22 Z^{8}+13 Z^{7}-6 Z^{6}-14 Z^{5}-6 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =1\right)^{-n +4}}{3641177375435009}-\frac{1463038124051290 \mathit{RootOf} \left(10 Z^{9}+22 Z^{8}+13 Z^{7}-6 Z^{6}-14 Z^{5}-6 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =2\right)^{-n +4}}{3641177375435009}-\frac{1463038124051290 \mathit{RootOf} \left(10 Z^{9}+22 Z^{8}+13 Z^{7}-6 Z^{6}-14 Z^{5}-6 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =3\right)^{-n +4}}{3641177375435009}-\frac{1463038124051290 \mathit{RootOf} \left(10 Z^{9}+22 Z^{8}+13 Z^{7}-6 Z^{6}-14 Z^{5}-6 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =4\right)^{-n +4}}{3641177375435009}-\frac{1463038124051290 \mathit{RootOf} \left(10 Z^{9}+22 Z^{8}+13 Z^{7}-6 Z^{6}-14 Z^{5}-6 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =5\right)^{-n +4}}{3641177375435009}-\frac{1463038124051290 \mathit{RootOf} \left(10 Z^{9}+22 Z^{8}+13 Z^{7}-6 Z^{6}-14 Z^{5}-6 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =6\right)^{-n +4}}{3641177375435009}-\frac{1463038124051290 \mathit{RootOf} \left(10 Z^{9}+22 Z^{8}+13 Z^{7}-6 Z^{6}-14 Z^{5}-6 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =7\right)^{-n +4}}{3641177375435009}-\frac{1463038124051290 \mathit{RootOf} \left(10 Z^{9}+22 Z^{8}+13 Z^{7}-6 Z^{6}-14 Z^{5}-6 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =8\right)^{-n +4}}{3641177375435009}-\frac{1463038124051290 \mathit{RootOf} \left(10 Z^{9}+22 Z^{8}+13 Z^{7}-6 Z^{6}-14 Z^{5}-6 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =9\right)^{-n +4}}{3641177375435009}-\frac{853855944516550 \mathit{RootOf} \left(10 Z^{9}+22 Z^{8}+13 Z^{7}-6 Z^{6}-14 Z^{5}-6 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =1\right)^{-n +3}}{3641177375435009}-\frac{853855944516550 \mathit{RootOf} \left(10 Z^{9}+22 Z^{8}+13 Z^{7}-6 Z^{6}-14 Z^{5}-6 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =2\right)^{-n +3}}{3641177375435009}-\frac{853855944516550 \mathit{RootOf} \left(10 Z^{9}+22 Z^{8}+13 Z^{7}-6 Z^{6}-14 Z^{5}-6 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =3\right)^{-n +3}}{3641177375435009}-\frac{853855944516550 \mathit{RootOf} \left(10 Z^{9}+22 Z^{8}+13 Z^{7}-6 Z^{6}-14 Z^{5}-6 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =4\right)^{-n +3}}{3641177375435009}-\frac{853855944516550 \mathit{RootOf} \left(10 Z^{9}+22 Z^{8}+13 Z^{7}-6 Z^{6}-14 Z^{5}-6 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =5\right)^{-n +3}}{3641177375435009}-\frac{853855944516550 \mathit{RootOf} \left(10 Z^{9}+22 Z^{8}+13 Z^{7}-6 Z^{6}-14 Z^{5}-6 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =6\right)^{-n +3}}{3641177375435009}-\frac{853855944516550 \mathit{RootOf} \left(10 Z^{9}+22 Z^{8}+13 Z^{7}-6 Z^{6}-14 Z^{5}-6 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =7\right)^{-n +3}}{3641177375435009}-\frac{853855944516550 \mathit{RootOf} \left(10 Z^{9}+22 Z^{8}+13 Z^{7}-6 Z^{6}-14 Z^{5}-6 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =8\right)^{-n +3}}{3641177375435009}-\frac{853855944516550 \mathit{RootOf} \left(10 Z^{9}+22 Z^{8}+13 Z^{7}-6 Z^{6}-14 Z^{5}-6 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =9\right)^{-n +3}}{3641177375435009}-\frac{99619254026685 \mathit{RootOf} \left(10 Z^{9}+22 Z^{8}+13 Z^{7}-6 Z^{6}-14 Z^{5}-6 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =1\right)^{-n +2}}{3641177375435009}-\frac{99619254026685 \mathit{RootOf} \left(10 Z^{9}+22 Z^{8}+13 Z^{7}-6 Z^{6}-14 Z^{5}-6 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =2\right)^{-n +2}}{3641177375435009}-\frac{99619254026685 \mathit{RootOf} \left(10 Z^{9}+22 Z^{8}+13 Z^{7}-6 Z^{6}-14 Z^{5}-6 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =3\right)^{-n +2}}{3641177375435009}-\frac{99619254026685 \mathit{RootOf} \left(10 Z^{9}+22 Z^{8}+13 Z^{7}-6 Z^{6}-14 Z^{5}-6 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =4\right)^{-n +2}}{3641177375435009}-\frac{99619254026685 \mathit{RootOf} \left(10 Z^{9}+22 Z^{8}+13 Z^{7}-6 Z^{6}-14 Z^{5}-6 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =5\right)^{-n +2}}{3641177375435009}-\frac{99619254026685 \mathit{RootOf} \left(10 Z^{9}+22 Z^{8}+13 Z^{7}-6 Z^{6}-14 Z^{5}-6 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =6\right)^{-n +2}}{3641177375435009}-\frac{99619254026685 \mathit{RootOf} \left(10 Z^{9}+22 Z^{8}+13 Z^{7}-6 Z^{6}-14 Z^{5}-6 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =7\right)^{-n +2}}{3641177375435009}-\frac{99619254026685 \mathit{RootOf} \left(10 Z^{9}+22 Z^{8}+13 Z^{7}-6 Z^{6}-14 Z^{5}-6 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =8\right)^{-n +2}}{3641177375435009}-\frac{99619254026685 \mathit{RootOf} \left(10 Z^{9}+22 Z^{8}+13 Z^{7}-6 Z^{6}-14 Z^{5}-6 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =9\right)^{-n +2}}{3641177375435009}+\frac{510172101850346 \mathit{RootOf} \left(10 Z^{9}+22 Z^{8}+13 Z^{7}-6 Z^{6}-14 Z^{5}-6 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =1\right)^{-n +1}}{3641177375435009}+\frac{510172101850346 \mathit{RootOf} \left(10 Z^{9}+22 Z^{8}+13 Z^{7}-6 Z^{6}-14 Z^{5}-6 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =2\right)^{-n +1}}{3641177375435009}+\frac{510172101850346 \mathit{RootOf} \left(10 Z^{9}+22 Z^{8}+13 Z^{7}-6 Z^{6}-14 Z^{5}-6 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =3\right)^{-n +1}}{3641177375435009}+\frac{510172101850346 \mathit{RootOf} \left(10 Z^{9}+22 Z^{8}+13 Z^{7}-6 Z^{6}-14 Z^{5}-6 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =4\right)^{-n +1}}{3641177375435009}+\frac{510172101850346 \mathit{RootOf} \left(10 Z^{9}+22 Z^{8}+13 Z^{7}-6 Z^{6}-14 Z^{5}-6 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =5\right)^{-n +1}}{3641177375435009}+\frac{510172101850346 \mathit{RootOf} \left(10 Z^{9}+22 Z^{8}+13 Z^{7}-6 Z^{6}-14 Z^{5}-6 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =6\right)^{-n +1}}{3641177375435009}+\frac{510172101850346 \mathit{RootOf} \left(10 Z^{9}+22 Z^{8}+13 Z^{7}-6 Z^{6}-14 Z^{5}-6 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =7\right)^{-n +1}}{3641177375435009}+\frac{510172101850346 \mathit{RootOf} \left(10 Z^{9}+22 Z^{8}+13 Z^{7}-6 Z^{6}-14 Z^{5}-6 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =8\right)^{-n +1}}{3641177375435009}+\frac{510172101850346 \mathit{RootOf} \left(10 Z^{9}+22 Z^{8}+13 Z^{7}-6 Z^{6}-14 Z^{5}-6 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =9\right)^{-n +1}}{3641177375435009}+\frac{339301805006260 \mathit{RootOf} \left(10 Z^{9}+22 Z^{8}+13 Z^{7}-6 Z^{6}-14 Z^{5}-6 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =1\right)^{-n -1}}{3641177375435009}+\frac{339301805006260 \mathit{RootOf} \left(10 Z^{9}+22 Z^{8}+13 Z^{7}-6 Z^{6}-14 Z^{5}-6 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =2\right)^{-n -1}}{3641177375435009}+\frac{339301805006260 \mathit{RootOf} \left(10 Z^{9}+22 Z^{8}+13 Z^{7}-6 Z^{6}-14 Z^{5}-6 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =3\right)^{-n -1}}{3641177375435009}+\frac{339301805006260 \mathit{RootOf} \left(10 Z^{9}+22 Z^{8}+13 Z^{7}-6 Z^{6}-14 Z^{5}-6 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =4\right)^{-n -1}}{3641177375435009}+\frac{339301805006260 \mathit{RootOf} \left(10 Z^{9}+22 Z^{8}+13 Z^{7}-6 Z^{6}-14 Z^{5}-6 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =5\right)^{-n -1}}{3641177375435009}+\frac{339301805006260 \mathit{RootOf} \left(10 Z^{9}+22 Z^{8}+13 Z^{7}-6 Z^{6}-14 Z^{5}-6 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =6\right)^{-n -1}}{3641177375435009}+\frac{339301805006260 \mathit{RootOf} \left(10 Z^{9}+22 Z^{8}+13 Z^{7}-6 Z^{6}-14 Z^{5}-6 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =7\right)^{-n -1}}{3641177375435009}+\frac{339301805006260 \mathit{RootOf} \left(10 Z^{9}+22 Z^{8}+13 Z^{7}-6 Z^{6}-14 Z^{5}-6 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =8\right)^{-n -1}}{3641177375435009}+\frac{339301805006260 \mathit{RootOf} \left(10 Z^{9}+22 Z^{8}+13 Z^{7}-6 Z^{6}-14 Z^{5}-6 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =9\right)^{-n -1}}{3641177375435009}+\frac{204991009875739 \mathit{RootOf} \left(10 Z^{9}+22 Z^{8}+13 Z^{7}-6 Z^{6}-14 Z^{5}-6 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =1\right)^{-n}}{3641177375435009}+\frac{204991009875739 \mathit{RootOf} \left(10 Z^{9}+22 Z^{8}+13 Z^{7}-6 Z^{6}-14 Z^{5}-6 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =2\right)^{-n}}{3641177375435009}+\frac{204991009875739 \mathit{RootOf} \left(10 Z^{9}+22 Z^{8}+13 Z^{7}-6 Z^{6}-14 Z^{5}-6 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =3\right)^{-n}}{3641177375435009}+\frac{204991009875739 \mathit{RootOf} \left(10 Z^{9}+22 Z^{8}+13 Z^{7}-6 Z^{6}-14 Z^{5}-6 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =4\right)^{-n}}{3641177375435009}+\frac{204991009875739 \mathit{RootOf} \left(10 Z^{9}+22 Z^{8}+13 Z^{7}-6 Z^{6}-14 Z^{5}-6 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =5\right)^{-n}}{3641177375435009}+\frac{204991009875739 \mathit{RootOf} \left(10 Z^{9}+22 Z^{8}+13 Z^{7}-6 Z^{6}-14 Z^{5}-6 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =6\right)^{-n}}{3641177375435009}+\frac{204991009875739 \mathit{RootOf} \left(10 Z^{9}+22 Z^{8}+13 Z^{7}-6 Z^{6}-14 Z^{5}-6 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =7\right)^{-n}}{3641177375435009}+\frac{204991009875739 \mathit{RootOf} \left(10 Z^{9}+22 Z^{8}+13 Z^{7}-6 Z^{6}-14 Z^{5}-6 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =8\right)^{-n}}{3641177375435009}+\frac{204991009875739 \mathit{RootOf} \left(10 Z^{9}+22 Z^{8}+13 Z^{7}-6 Z^{6}-14 Z^{5}-6 Z^{4}-4 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =9\right)^{-n}}{3641177375435009}\)
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This specification was found using the strategy pack "Point Placements" and has 123 rules.

Found on January 18, 2022.

Finding the specification took 3 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_{20}\! \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_{14}\! \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_{13}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{17}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{16}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{13}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)\\ F_{18}\! \left(x \right) &= F_{19}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{19}\! \left(x \right) &= F_{11}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{21}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{104}\! \left(x \right)+F_{22}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{22}\! \left(x \right) &= 0\\ F_{23}\! \left(x \right) &= F_{24}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{24}\! \left(x \right) &= F_{25}\! \left(x \right)+F_{38}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{26}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{27}\! \left(x \right)+F_{33}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{29}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{30}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{32}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{4}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{34}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{14}\! \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_{4}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{54}\! \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_{48}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{43}\! \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_{2}\! \left(x \right)+F_{46}\! \left(x \right)\\ F_{46}\! \left(x \right) &= F_{47}\! \left(x \right)\\ F_{47}\! \left(x \right) &= F_{2}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{48}\! \left(x \right) &= F_{49}\! \left(x \right)+F_{51}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{50}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{4}\! \left(x \right) F_{45}\! \left(x \right)\\ F_{51}\! \left(x \right) &= F_{52}\! \left(x \right)\\ F_{52}\! \left(x \right) &= F_{4}\! \left(x \right) F_{53}\! \left(x \right)\\ F_{53}\! \left(x \right) &= F_{43}\! \left(x \right)\\ F_{54}\! \left(x \right) &= 2 F_{22}\! \left(x \right)+F_{55}\! \left(x \right)+F_{97}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{4}\! \left(x \right) F_{56}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{57}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{43}\! \left(x \right)+F_{58}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{59}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{4}\! \left(x \right) F_{60}\! \left(x \right)\\ F_{60}\! \left(x \right) &= F_{61}\! \left(x \right)\\ F_{61}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{47}\! \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_{64}\! \left(x \right)+F_{72}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{65}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{66}\! \left(x \right)+F_{71}\! \left(x \right)\\ F_{66}\! \left(x \right) &= F_{4}\! \left(x \right) F_{67}\! \left(x \right)\\ F_{67}\! \left(x \right) &= F_{68}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{69}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{70}\! \left(x \right)\\ F_{70}\! \left(x \right) &= x^{2}\\ F_{71}\! \left(x \right) &= F_{15}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{73}\! \left(x \right)\\ F_{73}\! \left(x \right) &= 2 F_{22}\! \left(x \right)+F_{74}\! \left(x \right)+F_{79}\! \left(x \right)\\ F_{74}\! \left(x \right) &= F_{4}\! \left(x \right) F_{75}\! \left(x \right)\\ F_{75}\! \left(x \right) &= F_{76}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{46}\! \left(x \right)+F_{77}\! \left(x \right)\\ F_{77}\! \left(x \right) &= F_{78}\! \left(x \right)\\ F_{78}\! \left(x \right) &= F_{4}\! \left(x \right) F_{61}\! \left(x \right)\\ F_{79}\! \left(x \right) &= F_{4}\! \left(x \right) F_{80}\! \left(x \right)\\ F_{80}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{81}\! \left(x \right)+F_{95}\! \left(x \right)\\ F_{81}\! \left(x \right) &= F_{4}\! \left(x \right) F_{82}\! \left(x \right)\\ F_{82}\! \left(x \right) &= F_{83}\! \left(x \right)+F_{88}\! \left(x \right)\\ F_{83}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{84}\! \left(x \right)\\ F_{84}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{66}\! \left(x \right)+F_{85}\! \left(x \right)\\ F_{85}\! \left(x \right) &= F_{4}\! \left(x \right) F_{86}\! \left(x \right)\\ F_{86}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{87}\! \left(x \right)\\ F_{87}\! \left(x \right) &= F_{36}\! \left(x \right)\\ F_{88}\! \left(x \right) &= F_{49}\! \left(x \right)+F_{89}\! \left(x \right)\\ F_{89}\! \left(x \right) &= 2 F_{22}\! \left(x \right)+F_{74}\! \left(x \right)+F_{90}\! \left(x \right)\\ F_{90}\! \left(x \right) &= F_{4}\! \left(x \right) F_{91}\! \left(x \right)\\ F_{91}\! \left(x \right) &= F_{80}\! \left(x \right)+F_{92}\! \left(x \right)\\ F_{92}\! \left(x \right) &= F_{93}\! \left(x \right)\\ F_{93}\! \left(x \right) &= F_{4}\! \left(x \right) F_{94}\! \left(x \right)\\ F_{94}\! \left(x \right) &= F_{46}\! \left(x \right)\\ F_{95}\! \left(x \right) &= F_{4}\! \left(x \right) F_{96}\! \left(x \right)\\ F_{96}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{61}\! \left(x \right)\\ F_{97}\! \left(x \right) &= F_{4}\! \left(x \right) F_{98}\! \left(x \right)\\ F_{98}\! \left(x \right) &= F_{103}\! \left(x \right)+F_{99}\! \left(x \right)\\ F_{99}\! \left(x \right) &= F_{100}\! \left(x \right)+F_{80}\! \left(x \right)\\ F_{100}\! \left(x \right) &= F_{101}\! \left(x \right)\\ F_{101}\! \left(x \right) &= F_{102}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{102}\! \left(x \right) &= F_{43}\! \left(x \right)\\ F_{103}\! \left(x \right) &= F_{93}\! \left(x \right)\\ F_{104}\! \left(x \right) &= F_{105}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{105}\! \left(x \right) &= F_{106}\! \left(x \right)+F_{117}\! \left(x \right)\\ F_{106}\! \left(x \right) &= F_{107}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{107}\! \left(x \right) &= F_{108}\! \left(x \right)+F_{116}\! \left(x \right)+F_{22}\! \left(x \right)\\ F_{108}\! \left(x \right) &= F_{109}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{109}\! \left(x \right) &= F_{110}\! \left(x \right)+F_{113}\! \left(x \right)\\ F_{110}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{111}\! \left(x \right)\\ F_{111}\! \left(x \right) &= F_{112}\! \left(x \right)+F_{22}\! \left(x \right)+F_{27}\! \left(x \right)\\ F_{112}\! \left(x \right) &= F_{4}\! \left(x \right) F_{86}\! \left(x \right)\\ F_{113}\! \left(x \right) &= F_{114}\! \left(x \right)+F_{43}\! \left(x \right)\\ F_{114}\! \left(x \right) &= 2 F_{22}\! \left(x \right)+F_{115}\! \left(x \right)+F_{55}\! \left(x \right)\\ F_{115}\! \left(x \right) &= F_{4}\! \left(x \right) F_{91}\! \left(x \right)\\ F_{116}\! \left(x \right) &= F_{4}\! \left(x \right) F_{96}\! \left(x \right)\\ F_{117}\! \left(x \right) &= F_{118}\! \left(x \right)+F_{80}\! \left(x \right)\\ F_{118}\! \left(x \right) &= 2 F_{22}\! \left(x \right)+F_{101}\! \left(x \right)+F_{119}\! \left(x \right)\\ F_{119}\! \left(x \right) &= F_{120}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{120}\! \left(x \right) &= F_{121}\! \left(x \right)+F_{122}\! \left(x \right)\\ F_{121}\! \left(x \right) &= F_{17}\! \left(x \right)\\ F_{122}\! \left(x \right) &= F_{51}\! \left(x \right)\\ \end{align*}\)