Av(1234, 1243, 1324, 1432, 3214)
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Generating Function
\(\displaystyle -\frac{x^{4}+3 x^{3}+x^{2}-1}{4 x^{10}+16 x^{9}+17 x^{8}+6 x^{7}-7 x^{6}-9 x^{5}-5 x^{4}-5 x^{3}-2 x^{2}-x +1}\)
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Counting Sequence
1, 1, 2, 6, 19, 55, 149, 403, 1116, 3110, 8640, 23929, 66252, 183568, 508812, ...
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Implicit Equation for the Generating Function
\(\displaystyle \left(4 x^{10}+16 x^{9}+17 x^{8}+6 x^{7}-7 x^{6}-9 x^{5}-5 x^{4}-5 x^{3}-2 x^{2}-x +1\right) F \! \left(x \right)+x^{4}+3 x^{3}+x^{2}-1 = 0\)
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Recurrence
\(\displaystyle a \! \left(0\right) = 1\)
\(\displaystyle a \! \left(1\right) = 1\)
\(\displaystyle a \! \left(2\right) = 2\)
\(\displaystyle a \! \left(3\right) = 6\)
\(\displaystyle a \! \left(4\right) = 19\)
\(\displaystyle a \! \left(5\right) = 55\)
\(\displaystyle a \! \left(6\right) = 149\)
\(\displaystyle a \! \left(7\right) = 403\)
\(\displaystyle a \! \left(8\right) = 1116\)
\(\displaystyle a \! \left(9\right) = 3110\)
\(\displaystyle a \! \left(n +10\right) = -4 a \! \left(n \right)-16 a \! \left(n +1\right)-17 a \! \left(n +2\right)-6 a \! \left(n +3\right)+7 a \! \left(n +4\right)+9 a \! \left(n +5\right)+5 a \! \left(n +6\right)+5 a \! \left(n +7\right)+2 a \! \left(n +8\right)+a \! \left(n +9\right), \quad n \geq 10\)
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Explicit Closed Form
\(\displaystyle -\frac{32207991978230324 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =1\right)^{-n +8}}{155258632311136045}-\frac{32207991978230324 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =2\right)^{-n +8}}{155258632311136045}-\frac{32207991978230324 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =3\right)^{-n +8}}{155258632311136045}-\frac{32207991978230324 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =4\right)^{-n +8}}{155258632311136045}-\frac{32207991978230324 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =5\right)^{-n +8}}{155258632311136045}-\frac{32207991978230324 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =6\right)^{-n +8}}{155258632311136045}-\frac{32207991978230324 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =7\right)^{-n +8}}{155258632311136045}-\frac{32207991978230324 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =8\right)^{-n +8}}{155258632311136045}-\frac{32207991978230324 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =9\right)^{-n +8}}{155258632311136045}-\frac{32207991978230324 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =10\right)^{-n +8}}{155258632311136045}-\frac{22167685197394040 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =1\right)^{-n +7}}{31051726462227209}-\frac{22167685197394040 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =2\right)^{-n +7}}{31051726462227209}-\frac{22167685197394040 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =3\right)^{-n +7}}{31051726462227209}-\frac{22167685197394040 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =4\right)^{-n +7}}{31051726462227209}-\frac{22167685197394040 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =5\right)^{-n +7}}{31051726462227209}-\frac{22167685197394040 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =6\right)^{-n +7}}{31051726462227209}-\frac{22167685197394040 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =7\right)^{-n +7}}{31051726462227209}-\frac{22167685197394040 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =8\right)^{-n +7}}{31051726462227209}-\frac{22167685197394040 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =9\right)^{-n +7}}{31051726462227209}-\frac{22167685197394040 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =10\right)^{-n +7}}{31051726462227209}-\frac{64948856793633577 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =1\right)^{-n +6}}{155258632311136045}-\frac{64948856793633577 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =2\right)^{-n +6}}{155258632311136045}-\frac{64948856793633577 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =3\right)^{-n +6}}{155258632311136045}-\frac{64948856793633577 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =4\right)^{-n +6}}{155258632311136045}-\frac{64948856793633577 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =5\right)^{-n +6}}{155258632311136045}-\frac{64948856793633577 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =6\right)^{-n +6}}{155258632311136045}-\frac{64948856793633577 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =7\right)^{-n +6}}{155258632311136045}-\frac{64948856793633577 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =8\right)^{-n +6}}{155258632311136045}-\frac{64948856793633577 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =9\right)^{-n +6}}{155258632311136045}-\frac{64948856793633577 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =10\right)^{-n +6}}{155258632311136045}+\frac{10908512902856352 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =1\right)^{-n +5}}{155258632311136045}+\frac{10908512902856352 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =2\right)^{-n +5}}{155258632311136045}+\frac{10908512902856352 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =3\right)^{-n +5}}{155258632311136045}+\frac{10908512902856352 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =4\right)^{-n +5}}{155258632311136045}+\frac{10908512902856352 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =5\right)^{-n +5}}{155258632311136045}+\frac{10908512902856352 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =6\right)^{-n +5}}{155258632311136045}+\frac{10908512902856352 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =7\right)^{-n +5}}{155258632311136045}+\frac{10908512902856352 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =8\right)^{-n +5}}{155258632311136045}+\frac{10908512902856352 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =9\right)^{-n +5}}{155258632311136045}+\frac{10908512902856352 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =10\right)^{-n +5}}{155258632311136045}+\frac{26233201813983284 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =1\right)^{-n +4}}{155258632311136045}+\frac{26233201813983284 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =2\right)^{-n +4}}{155258632311136045}+\frac{26233201813983284 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =3\right)^{-n +4}}{155258632311136045}+\frac{26233201813983284 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =4\right)^{-n +4}}{155258632311136045}+\frac{26233201813983284 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =5\right)^{-n +4}}{155258632311136045}+\frac{26233201813983284 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =6\right)^{-n +4}}{155258632311136045}+\frac{26233201813983284 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =7\right)^{-n +4}}{155258632311136045}+\frac{26233201813983284 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =8\right)^{-n +4}}{155258632311136045}+\frac{26233201813983284 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =9\right)^{-n +4}}{155258632311136045}+\frac{26233201813983284 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =10\right)^{-n +4}}{155258632311136045}+\frac{14175593626436033 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =1\right)^{-n +3}}{155258632311136045}+\frac{14175593626436033 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =2\right)^{-n +3}}{155258632311136045}+\frac{14175593626436033 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =3\right)^{-n +3}}{155258632311136045}+\frac{14175593626436033 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =4\right)^{-n +3}}{155258632311136045}+\frac{14175593626436033 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =5\right)^{-n +3}}{155258632311136045}+\frac{14175593626436033 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =6\right)^{-n +3}}{155258632311136045}+\frac{14175593626436033 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =7\right)^{-n +3}}{155258632311136045}+\frac{14175593626436033 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =8\right)^{-n +3}}{155258632311136045}+\frac{14175593626436033 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =9\right)^{-n +3}}{155258632311136045}+\frac{14175593626436033 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =10\right)^{-n +3}}{155258632311136045}+\frac{12740678177383828 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =1\right)^{-n +2}}{155258632311136045}+\frac{12740678177383828 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =2\right)^{-n +2}}{155258632311136045}+\frac{12740678177383828 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =3\right)^{-n +2}}{155258632311136045}+\frac{12740678177383828 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =4\right)^{-n +2}}{155258632311136045}+\frac{12740678177383828 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =5\right)^{-n +2}}{155258632311136045}+\frac{12740678177383828 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =6\right)^{-n +2}}{155258632311136045}+\frac{12740678177383828 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =7\right)^{-n +2}}{155258632311136045}+\frac{12740678177383828 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =8\right)^{-n +2}}{155258632311136045}+\frac{12740678177383828 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =9\right)^{-n +2}}{155258632311136045}+\frac{12740678177383828 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =10\right)^{-n +2}}{155258632311136045}+\frac{38838788152470408 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =1\right)^{-n +1}}{155258632311136045}+\frac{38838788152470408 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =2\right)^{-n +1}}{155258632311136045}+\frac{38838788152470408 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =3\right)^{-n +1}}{155258632311136045}+\frac{38838788152470408 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =4\right)^{-n +1}}{155258632311136045}+\frac{38838788152470408 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =5\right)^{-n +1}}{155258632311136045}+\frac{38838788152470408 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =6\right)^{-n +1}}{155258632311136045}+\frac{38838788152470408 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =7\right)^{-n +1}}{155258632311136045}+\frac{38838788152470408 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =8\right)^{-n +1}}{155258632311136045}+\frac{38838788152470408 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =9\right)^{-n +1}}{155258632311136045}+\frac{38838788152470408 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =10\right)^{-n +1}}{155258632311136045}+\frac{4551243551595971 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =1\right)^{-n -1}}{155258632311136045}+\frac{4551243551595971 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =2\right)^{-n -1}}{155258632311136045}+\frac{4551243551595971 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =3\right)^{-n -1}}{155258632311136045}+\frac{4551243551595971 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =4\right)^{-n -1}}{155258632311136045}+\frac{4551243551595971 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =5\right)^{-n -1}}{155258632311136045}+\frac{4551243551595971 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =6\right)^{-n -1}}{155258632311136045}+\frac{4551243551595971 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =7\right)^{-n -1}}{155258632311136045}+\frac{4551243551595971 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =8\right)^{-n -1}}{155258632311136045}+\frac{4551243551595971 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =9\right)^{-n -1}}{155258632311136045}+\frac{4551243551595971 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =10\right)^{-n -1}}{155258632311136045}+\frac{4182254103515372 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =6\right)^{-n}}{31051726462227209}+\frac{4182254103515372 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =7\right)^{-n}}{31051726462227209}+\frac{4182254103515372 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =8\right)^{-n}}{31051726462227209}+\frac{4182254103515372 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =9\right)^{-n}}{31051726462227209}+\frac{4182254103515372 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =10\right)^{-n}}{31051726462227209}+\frac{4182254103515372 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =1\right)^{-n}}{31051726462227209}+\frac{4182254103515372 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =2\right)^{-n}}{31051726462227209}+\frac{4182254103515372 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =3\right)^{-n}}{31051726462227209}+\frac{4182254103515372 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =4\right)^{-n}}{31051726462227209}+\frac{4182254103515372 \mathit{RootOf} \left(4 Z^{10}+16 Z^{9}+17 Z^{8}+6 Z^{7}-7 Z^{6}-9 Z^{5}-5 Z^{4}-5 Z^{3}-2 Z^{2}-Z +1, \mathit{index} =5\right)^{-n}}{31051726462227209}\)
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This specification was found using the strategy pack "Point Placements" and has 89 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_{16}\! \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_{11}\! \left(x \right)\\ F_{10}\! \left(x \right) &= F_{1}\! \left(x \right)+F_{4}\! \left(x \right)\\ F_{11}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{14}\! \left(x \right)\\ F_{12}\! \left(x \right) &= F_{13}\! \left(x \right)\\ F_{13}\! \left(x \right) &= F_{10}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{14}\! \left(x \right) &= F_{15}\! \left(x \right)\\ F_{15}\! \left(x \right) &= F_{12}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{16}\! \left(x \right) &= F_{17}\! \left(x \right)+F_{2}\! \left(x \right)\\ F_{17}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{19}\! \left(x \right)+F_{79}\! \left(x \right)\\ F_{18}\! \left(x \right) &= 0\\ F_{19}\! \left(x \right) &= F_{20}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{20}\! \left(x \right) &= F_{21}\! \left(x \right)+F_{34}\! \left(x \right)\\ F_{21}\! \left(x \right) &= F_{22}\! \left(x \right)+F_{7}\! \left(x \right)\\ F_{22}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{23}\! \left(x \right)+F_{27}\! \left(x \right)\\ 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_{26}\! \left(x \right)\\ F_{25}\! \left(x \right) &= F_{4}\! \left(x \right)\\ F_{26}\! \left(x \right) &= F_{14}\! \left(x \right)\\ F_{27}\! \left(x \right) &= F_{28}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{28}\! \left(x \right) &= F_{11}\! \left(x \right)+F_{29}\! \left(x \right)\\ F_{29}\! \left(x \right) &= F_{30}\! \left(x \right)+F_{32}\! \left(x \right)\\ F_{30}\! \left(x \right) &= F_{31}\! \left(x \right)\\ F_{31}\! \left(x \right) &= F_{25}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{32}\! \left(x \right) &= F_{33}\! \left(x \right)\\ F_{33}\! \left(x \right) &= F_{30}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{34}\! \left(x \right) &= F_{35}\! \left(x \right)+F_{46}\! \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_{38}\! \left(x \right)+F_{41}\! \left(x \right)\\ F_{38}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{39}\! \left(x \right)\\ F_{39}\! \left(x \right) &= F_{40}\! \left(x \right)\\ F_{40}\! \left(x \right) &= F_{2}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{41}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{44}\! \left(x \right)\\ F_{42}\! \left(x \right) &= F_{43}\! \left(x \right)\\ F_{43}\! \left(x \right) &= F_{38}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{44}\! \left(x \right) &= F_{45}\! \left(x \right)\\ F_{45}\! \left(x \right) &= F_{4}\! \left(x \right) F_{42}\! \left(x \right)\\ F_{46}\! \left(x \right) &= 2 F_{18}\! \left(x \right)+F_{47}\! \left(x \right)+F_{72}\! \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_{50}\! \left(x \right)\\ F_{49}\! \left(x \right) &= F_{39}\! \left(x \right)\\ F_{50}\! \left(x \right) &= F_{51}\! \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_{18}\! \left(x \right)+F_{54}\! \left(x \right)+F_{63}\! \left(x \right)\\ F_{54}\! \left(x \right) &= F_{4}\! \left(x \right) F_{55}\! \left(x \right)\\ F_{55}\! \left(x \right) &= F_{56}\! \left(x \right)+F_{59}\! \left(x \right)\\ F_{56}\! \left(x \right) &= F_{12}\! \left(x \right)+F_{57}\! \left(x \right)\\ F_{57}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{23}\! \left(x \right)+F_{58}\! \left(x \right)\\ F_{58}\! \left(x \right) &= F_{11}\! \left(x \right) F_{4}\! \left(x \right)\\ F_{59}\! \left(x \right) &= F_{42}\! \left(x \right)+F_{60}\! \left(x \right)\\ F_{60}\! \left(x \right) &= 2 F_{18}\! \left(x \right)+F_{47}\! \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_{51}\! \left(x \right)+F_{53}\! \left(x \right)\\ F_{63}\! \left(x \right) &= F_{4}\! \left(x \right) F_{64}\! \left(x \right)\\ F_{64}\! \left(x \right) &= F_{2}\! \left(x \right)+F_{65}\! \left(x \right)\\ F_{65}\! \left(x \right) &= F_{18}\! \left(x \right)+F_{40}\! \left(x \right)+F_{66}\! \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_{70}\! \left(x \right)\\ F_{68}\! \left(x \right) &= F_{4}\! \left(x \right)+F_{69}\! \left(x \right)\\ F_{69}\! \left(x \right) &= F_{15}\! \left(x \right)+F_{18}\! \left(x \right)+F_{23}\! \left(x \right)\\ F_{70}\! \left(x \right) &= F_{39}\! \left(x \right)+F_{71}\! \left(x \right)\\ F_{71}\! \left(x \right) &= 2 F_{18}\! \left(x \right)+F_{47}\! \left(x \right)+F_{52}\! \left(x \right)\\ F_{72}\! \left(x \right) &= F_{4}\! \left(x \right) F_{73}\! \left(x \right)\\ F_{73}\! \left(x \right) &= F_{62}\! \left(x \right)+F_{74}\! \left(x \right)\\ F_{74}\! \left(x \right) &= F_{75}\! \left(x \right)+F_{77}\! \left(x \right)\\ F_{75}\! \left(x \right) &= F_{76}\! \left(x \right)\\ F_{76}\! \left(x \right) &= F_{4}\! \left(x \right) F_{49}\! \left(x \right)\\ F_{77}\! \left(x \right) &= F_{78}\! \left(x \right)\\ F_{78}\! \left(x \right) &= F_{4}\! \left(x \right) F_{75}\! \left(x \right)\\ F_{79}\! \left(x \right) &= F_{4}\! \left(x \right) F_{80}\! \left(x \right)\\ F_{80}\! \left(x \right) &= F_{64}\! \left(x \right)+F_{81}\! \left(x \right)\\ F_{81}\! \left(x \right) &= F_{53}\! \left(x \right)+F_{82}\! \left(x \right)\\ F_{82}\! \left(x \right) &= 2 F_{18}\! \left(x \right)+F_{52}\! \left(x \right)+F_{83}\! \left(x \right)\\ F_{83}\! \left(x \right) &= F_{4}\! \left(x \right) F_{84}\! \left(x \right)\\ F_{84}\! \left(x \right) &= F_{85}\! \left(x \right)+F_{87}\! \left(x \right)\\ F_{85}\! \left(x \right) &= F_{14}\! \left(x \right)+F_{86}\! \left(x \right)\\ F_{86}\! \left(x \right) &= F_{33}\! \left(x \right)\\ F_{87}\! \left(x \right) &= F_{44}\! \left(x \right)+F_{88}\! \left(x \right)\\ F_{88}\! \left(x \right) &= F_{78}\! \left(x \right)\\ \end{align*}\)