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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 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Reduction of a function f(x) by a substitution q(x)->s(x) is introduced at A193010.

LINKS

Table of n, a(n) for n=1..100.

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

Cf. A000045, A001622, A193010, A192232, A193032.

Sequence in context: A114195 A185948 A016601 * A078416 A223533 A021973

Adjacent sequences:  A193028 A193029 A193030 * A193032 A193033 A193034

KEYWORD

nonn,cons

AUTHOR

Clark Kimberling, Jul 14 2011

STATUS

approved

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Last modified May 25 09:37 EDT 2022. Contains 354066 sequences. (Running on oeis4.)