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A193029
Decimal expansion of the constant term of the reduction of e^(x/2) by x^2->x+1.
2
1, 1, 5, 1, 9, 4, 5, 2, 0, 6, 7, 5, 9, 4, 6, 8, 8, 2, 1, 2, 7, 7, 4, 8, 5, 4, 5, 1, 5, 0, 5, 5, 8, 2, 7, 4, 3, 2, 1, 2, 3, 8, 5, 8, 9, 0, 4, 1, 1, 3, 1, 5, 1, 1, 6, 6, 5, 2, 0, 0, 0, 1, 1, 8, 0, 1, 6, 4, 6, 0, 3, 2, 4, 0, 0, 6, 2, 0, 8, 2, 5, 1, 5, 5, 5, 1, 6, 3, 9, 7, 9, 8, 2, 7, 2, 7, 8, 1, 0, 0
OFFSET
1,3
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 1 + Sum_{k>=1} Fibonacci(k-1)/(k!*2^k).
Equals exp(1/4)*(cosh(sqrt(5)/4) - sqrt(5)*sinh(sqrt(5)/4)/5). (End)
EXAMPLE
1.151945206759468821277485451505582743212385890...
MATHEMATICA
f[x_] := Exp[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]
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Clark Kimberling, Jul 14 2011
STATUS
approved