A265292 Decimal expansion of Sum_{n >= 1} (c(2*n) - x), where c(n) = the n-th convergent to x = sqrt(2).

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

Offset

0,2

Links

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

Formula

From Peter Bala, Aug 23 2022: (Start)

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

Equals Sum_{n >= 1} 1/( (1 + sqrt(2))^(4*n) - 1).

A more rapidly converging series for the constant is 2*sqrt(2)*Sum_{n >= 1} x^(n^2)*(1 + x^n)/(1 - x^n), where x = 17 - 12*sqrt(2) = 0.029437.... See A000005. (End)

Example

sum = 0.0883138821525759032178529847253...

Maple

x := 17 - 12*sqrt(2) :
evalf(2*sqrt(2)*add( x^(n^2)*(1 + x^n)/(1 - x^n), n = 1..8), 100); # Peter Bala, Aug 23 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 *)

Crossrefs

Cf. A000005, A000129, A002193, A265291, A265293, A265288 (guide).

Sequence in context: A372776 A185266 A210630 * A272415 A010530 A289915

Adjacent sequences: A265289 A265290 A265291 * A265293 A265294 A265295

Keyword

nonn,cons

Author

Clark Kimberling, Dec 06 2015

Status

approved