cp's OEIS Frontend

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

%I A265292 #9 Aug 23 2022 09:52:53
%S A265292 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,
%T A265292 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,
%U A265292 2,9,2,7,4,5,2,6,3,7,7,2,6,4,7,0,4,3
%N A265292 Decimal expansion of Sum_{n >= 1} (c(2*n) - x), where c(n) = the n-th convergent to x = sqrt(2).
%F A265292 From _Peter Bala_, Aug 23 2022: (Start)
%F A265292 Equals Sum_{n >= 1} 1/( (1 + sqrt(2))^(2*n)*Pell(2*n) ), where Pell(n) = A000129(n).
%F A265292 Equals Sum_{n >= 1} 1/( (1 + sqrt(2))^(4*n) - 1).
%F A265292 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)
%e A265292 sum = 0.0883138821525759032178529847253...
%p A265292 x := 17 - 12*sqrt(2) :
%p A265292 evalf(2*sqrt(2)*add( x^(n^2)*(1 + x^n)/(1 - x^n), n = 1..8), 100); # _Peter Bala_, Aug 23 2022
%t A265292 x = Sqrt[2]; z = 600; c = Convergents[x, z];
%t A265292 s1 = Sum[x - c[[2 k - 1]], {k, 1, z/2}]; N[s1, 200]
%t A265292 s2 = Sum[c[[2 k]] - x, {k, 1, z/2}]; N[s2, 200]
%t A265292 N[s1 + s2, 200]
%t A265292 RealDigits[s1, 10, 120][[1]]  (* A265291 *)
%t A265292 RealDigits[s2, 10, 120][[1]]  (* A265292 *)
%t A265292 RealDigits[s1 + s2, 10, 120][[1]](* A265293 *)
%Y A265292 Cf. A000005, A000129, A002193, A265291, A265293, A265288 (guide).
%K A265292 nonn,cons
%O A265292 0,2
%A A265292 _Clark Kimberling_, Dec 06 2015