A265291 Decimal expansion of Sum_{n >= 1} (x - c(2n-1)), where c(n) = the n-th convergent to x = sqrt(2).

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

Offset

0,1

Links

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

Formula

From Peter Bala, Aug 20 2022: (Start)

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

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

Example

sum = 0.4288603380680959830021113676132723...

Maple

x := 2*sqrt(2) - 3:
evalf(2*sqrt(2)*add( x^(n*(n+1)/2)/(x^n - 1), n = 1..16), 100); # Peter Bala, Aug 21 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. A000129, A001227, A002193, A265292, A265293, A265288 (guide).

Sequence in context: A197016 A198145 A143942 * A195777 A125065 A109816

Adjacent sequences: A265288 A265289 A265290 * A265292 A265293 A265294

Keyword

nonn,cons

Author

Clark Kimberling, Dec 06 2015

Status

approved