A265292 - OEIS

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

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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 *)