A265296 Decimal expansion of Sum_{n >= 1} (c(2*n) - c(2*n-1)), where c(n) = the n-th convergent to x = sqrt(3).

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

Offset: 1

Author

Clark Kimberling, Dec 07 2015

Keywords

• nonn
• cons

Examples

			sum = 1.0898631708870032278893257211397258128825141977596999...
		

Crossrefs

Cf. A002194, A002530, A002531, A265294, A265295, A265288 (guide).

Programs

Maple

x := 2 - sqrt(3):
evalf(2*sqrt(3)*add(x^(n^2)*(1 + x^(2*n))/(1 - x^(2*n)), n = 1..13), 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 *)

Formula

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