A265293 Decimal expansion of Sum_{n >= 1} (c(2*n) - c(2*n-1)), where c(n) = the n-th convergent to x = sqrt(2).

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

Offset: 0

Author

Clark Kimberling, Dec 06 2015

Keywords

• nonn
• cons

Examples

			sum = 0.51717422022067188621996435233866923610552...
		

Crossrefs

Cf. A000129, A002193, A112329, A265291, A265292, A265288 (guide).

Programs

Maple

x := 3 - 2*sqrt(2):
evalf(2*sqrt(2)*add(x^(n^2)*(1 + x^(2*n))/(1 - x^(2*n)), n = 1..12), 100); # Peter Bala, Aug 20 2022

Mathematica

x = Sqrt[2]; 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]]  (* A265291 *)
RealDigits[s2, 10, 120][[1]]  (* A265292 *)
RealDigits[s1 + s2, 10, 120][[1]](* A265293 *)

Formula

From Peter Bala, Aug 20 2022: (Start)

Constant equals Sum_{n >= 1} 1/((1 + sqrt(2))^n*Pell(n)) = 2*sqrt(2)*Sum_{n >= 1} 1/( (3 + 2*sqrt(2))^n - (-1)^n ), where Pell(n) = A000129(n).

A more rapidly converging series for the constant is 2*sqrt(2)*Sum_{n >= 1} x^(n^2)*(1 + x^(2*n))/(1 - x^(2*n)), where x = 3 - 2*sqrt(2). See A112329. (End)