A265295 Decimal expansion of Sum_{n >= 1} (c(2*n) - x), where c(n) = the n-th convergent to x = sqrt(3).

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

Offset

0,1

Links

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

Formula

Equals 2*sqrt(3)*Sum_{n >= 1} x^(n^2)*(1 + x^n)/(1 - x^n), where x = 7 - 4*sqrt(3). - Peter Bala, Aug 24 2022

Example

sum = 0.28728008008348839351145153987668331682390...

Maple

x := 7 - 4*sqrt(3):
evalf(2*sqrt(3)*add( x^(n^2)*(1 + x^n)/(1 - x^n), n = 1..10), 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, A265294, A265296, A265288 (guide).

Sequence in context: A282089 A195367 A135725 * A093624 A021352 A094289

Adjacent sequences: A265292 A265293 A265294 * A265296 A265297 A265298

Keyword

nonn,cons

Author

Clark Kimberling, Dec 07 2015

Status

approved