A265293 - OEIS

%I #14 Aug 23 2022 09:53:00

%S 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,

%T 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,

%U 9,8,5,8,8,2,3,3,0,3,0,8,5,2,9,6,5,8

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

%F From _Peter Bala_, Aug 20 2022: (Start)

%F 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).

%F 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)

%e sum = 0.51717422022067188621996435233866923610552...

%p x := 3 - 2*sqrt(2):

%p 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

%t x = Sqrt[2]; z = 600; c = Convergents[x, z];

%t s1 = Sum[x - c[[2 k - 1]], {k, 1, z/2}]; N[s1, 200]

%t s2 = Sum[c[[2 k]] - x, {k, 1, z/2}]; N[s2, 200]

%t N[s1 + s2, 200]

%t RealDigits[s1, 10, 120][[1]]  (* A265291 *)

%t RealDigits[s2, 10, 120][[1]]  (* A265292 *)

%t RealDigits[s1 + s2, 10, 120][[1]](* A265293 *)

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

%K nonn,cons

%O 0,1

%A _Clark Kimberling_, Dec 06 2015