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A193088
Decimal expansion of the coefficient of x in the reduction of (cos(x))^2 by x^2->x+1 (negated).
3
2, 9, 6, 0, 6, 7, 2, 8, 1, 1, 2, 9, 8, 0, 8, 0, 5, 1, 0, 7, 9, 7, 3, 9, 7, 7, 0, 1, 3, 9, 6, 9, 6, 3, 9, 2, 9, 1, 6, 5, 2, 7, 5, 6, 4, 7, 6, 9, 2, 8, 5, 8, 3, 3, 3, 5, 0, 7, 5, 8, 0, 7, 9, 8, 7, 6, 2, 8, 9, 2, 5, 3, 0, 4, 5, 2, 5, 8, 2, 4, 8, 8, 3, 8, 4, 4, 3, 4, 7, 1, 8, 5, 1, 6, 5, 6, 5, 7, 3, 4
OFFSET
0,1
COMMENTS
Reduction of a function f(x) by a substitution q(x)->s(x) is introduced at A193010.
The positive value +0.2960... is the coefficient of x in the reduction of (sin(x))^2 by x^2->x+1.
FORMULA
From Amiram Eldar, Jan 19 2022: (Start)
Equals Sum_{k>=0} (-1)^k * 2^(2*k-1) * Fibonacci(2*k)/(2*k)!.
Equals -sin(1)*sin(sqrt(5))/sqrt(5). (End)
EXAMPLE
-0.29606728112980805107973977013969639291652...
MATHEMATICA
f[x_] := Cos[x]^2; r[n_] := Fibonacci[n];
c[n_] := SeriesCoefficient[Series[f[x], {x, 0, n}], n]
u1 = N[Sum[c[n]*r[n], {n, 0, 100}], 100]
RealDigits[u1, 10]
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Clark Kimberling, Jul 15 2011
STATUS
approved