A265294 Decimal expansion of Sum_{n>=1} (x - c(2n-1)), where c = convergents to (x = sqrt(3)).

8, 0, 2, 5, 8, 3, 0, 9, 0, 8, 0, 3, 5, 1, 4, 8, 3, 4, 3, 7, 7, 8, 7, 4, 1, 8, 1, 2, 6, 3, 0, 4, 2, 4, 9, 6, 0, 5, 8, 6, 0, 4, 7, 7, 6, 8, 9, 5, 1, 3, 2, 7, 7, 7, 0, 1, 7, 7, 4, 1, 8, 5, 1, 4, 8, 4, 1, 4, 0, 6, 0, 8, 4, 4, 8, 7, 0, 0, 3, 0, 2, 2, 1, 7, 9, 4

Offset

0,1

Links

Table of n, a(n) for n=0..85.

Formula

From Peter Bala, Aug 24 2022: (Start)

Equals 2*sqrt(3)*Sum_{n >= 1} 1/( 1 + (2+sqrt(3))^(2*n-1) ).

A more rapidly converging series for the constant is 2*sqrt(3)*Sum_{n >= 1} x^(n*(n+1)/2)/(x^n - 1), where x = sqrt(3) - 2. See A001227. (End)

Example

0.8025830908035148343778741812630...

Maple

x := sqrt(3) - 2:
evalf(2*sqrt(3)*add( x^(n*(n+1)/2)/(x^n - 1), n = 1..18), 100); # Peter Bala, Aug 24 2022

Mathematica

x = Sqrt[3]; z = 600; c = Convergents[x, z];
s1 = Sum[x - c[[2 k - 1]], {k, 1, z/2}]; N[s1, 200]
s2 = Sum[c[[2 k]] - x, {k, 1, z/2}]; N[s2, 200]
N[s1 + s2, 200]
RealDigits[s1, 10, 120][[1]]  (* A265294 *)
RealDigits[s2, 10, 120][[1]]  (* A265295 *)
RealDigits[s1 + s2, 10, 120][[1]]  (* A265296 *)

Crossrefs

Cf. A002194, A002530, A002531, A265295, A265296, A265288 (guide).

Sequence in context: A329086 A396739 A342200 * A062522 A117888 A331542

Adjacent sequences: A265291 A265292 A265293 * A265295 A265296 A265297

Keyword

nonn,cons

Author

Clark Kimberling, Dec 07 2015

Status

approved