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A193089
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Decimal expansion of the constant term of the reduction of (sin(x))^2 by x^2->x+1.
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2
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5, 1, 8, 7, 2, 3, 3, 3, 8, 6, 8, 4, 9, 2, 9, 1, 9, 6, 9, 9, 3, 0, 2, 7, 7, 7, 7, 0, 3, 8, 8, 6, 5, 6, 0, 3, 0, 4, 1, 5, 2, 8, 4, 0, 5, 3, 5, 4, 2, 4, 5, 3, 7, 9, 2, 7, 2, 3, 3, 3, 6, 9, 3, 7, 4, 6, 9, 9, 7, 2, 0, 6, 8, 7, 5, 2, 5, 5, 6, 0, 8, 3, 1, 2, 8, 2, 1, 0, 5, 5, 2, 4, 0, 1, 3, 1, 5, 4, 7, 0
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OFFSET
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0,1
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COMMENTS
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Reduction of a function f(x) by a substitution q(x)->s(x) is introduced at A193010.
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LINKS
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FORMULA
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Equals Sum_{k>=1} (-1)^(k+1) * 2^(2*k-1) * Fibonacci(2*k-1)/(2*k)!.
Equals (1 + 2*phi^2*sin(phi-1)^2 - cos(2*phi))/(5+sqrt(5)), where phi is the golden ratio (A001622). (End)
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EXAMPLE
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0.5187233386849291969930277770388656030...
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MATHEMATICA
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f[x_] := Sin[x]^2; r[n_] := Fibonacci[n];
c[n_] := SeriesCoefficient[Series[f[x], {x, 0, n}], n]
u0 = N[Sum[c[n]*r[n - 1], {n, 0, 100}], 100]
RealDigits[u0, 10]
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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