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A193031
Decimal expansion of the constant term of the reduction of 2^x by x^2->x+1.
2
1, 3, 1, 9, 8, 7, 8, 7, 2, 4, 0, 2, 1, 1, 5, 6, 2, 7, 4, 4, 5, 9, 9, 7, 4, 2, 1, 2, 6, 3, 9, 3, 1, 3, 9, 3, 1, 8, 5, 9, 0, 4, 4, 6, 2, 3, 0, 5, 5, 5, 9, 7, 8, 8, 1, 5, 1, 7, 5, 9, 4, 3, 2, 8, 8, 5, 3, 2, 2, 7, 6, 2, 6, 1, 5, 1, 5, 6, 0, 3, 7, 5, 5, 3, 5, 6, 1, 2, 4, 8, 2, 3, 1, 1, 3, 2, 2, 0, 2, 8
OFFSET
1,2
COMMENTS
Reduction of a function f(x) by a substitution q(x)->s(x) is introduced at A193010.
FORMULA
From Amiram Eldar, Jan 19 2022: (Start)
Equals 1 + Sum_{k>=1} log(2)^k*Fibonacci(k-1)/k!.
Equals (1 + (3+4^phi)/sqrt(5))/(phi*2^phi), where phi is the golden ratio (A001622). (End)
EXAMPLE
1.3198787240211562744599742126393139318590...
MATHEMATICA
f[x_] := 2^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, 100}], 100]
RealDigits[u0, 10]
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Clark Kimberling, Jul 14 2011
STATUS
approved