login
A193015
Decimal expansion of the constant term of the reduction of sinh(x) by x^2->x+1.
2
1, 9, 3, 3, 1, 3, 2, 3, 9, 9, 0, 6, 4, 1, 7, 3, 0, 4, 8, 7, 6, 4, 4, 3, 7, 9, 6, 0, 9, 0, 5, 8, 8, 0, 5, 5, 9, 9, 1, 9, 4, 4, 9, 3, 1, 3, 0, 7, 3, 5, 1, 4, 7, 2, 5, 7, 1, 4, 5, 8, 9, 2, 3, 2, 1, 5, 0, 7, 9, 1, 9, 0, 7, 6, 5, 1, 8, 5, 7, 4, 3, 0, 0, 5, 9, 2, 3, 9, 0, 4, 6, 3, 6, 6, 3, 9, 6, 0, 4, 6
OFFSET
0,2
COMMENTS
Reduction of a function f(x) by a substitution q(x)->s(x) is introduced at A193010.
FORMULA
From Amiram Eldar, Jan 18 2022: (Start)
Equals Sum_{k>=1} Fibonacci(2*k)/(2*k+1)!.
Equals (2 * sinh(phi) - (3+sqrt(5))*sinh(1/phi))/(5+sqrt(5)), where phi is the golden ratio (A001622). (End)
EXAMPLE
0.1933132399064173048764437960905880...
MATHEMATICA
f[x_] := Sinh[x]; r[n_] := Fibonacci[n];
c[n_] := SeriesCoefficient[Series[f[x], {x, 0, n}], n]
u0 = N[Sum[c[n]*r[n - 1], {n, 0, 300}], 100]
RealDigits[u0, 10]
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Clark Kimberling, Jul 14 2011
STATUS
approved