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A265293 Decimal expansion of Sum_{n >= 1} (c(2*n) - c(2*n-1)), where c(n) = the n-th convergent to x = sqrt(2).

%I A265293 #14 Aug 23 2022 09:53:00
%S A265293 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 A265293 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 A265293 9,8,5,8,8,2,3,3,0,3,0,8,5,2,9,6,5,8
%N A265293 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 A265293 From _Peter Bala_, Aug 20 2022: (Start)
%F A265293 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 A265293 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 A265293 sum = 0.51717422022067188621996435233866923610552...
%p A265293 x := 3 - 2*sqrt(2):
%p A265293 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 A265293 x = Sqrt[2]; z = 600; c = Convergents[x, z];
%t A265293 s1 = Sum[x - c[[2 k - 1]], {k, 1, z/2}]; N[s1, 200]
%t A265293 s2 = Sum[c[[2 k]] - x, {k, 1, z/2}]; N[s2, 200]
%t A265293 N[s1 + s2, 200]
%t A265293 RealDigits[s1, 10, 120][[1]]  (* A265291 *)
%t A265293 RealDigits[s2, 10, 120][[1]]  (* A265292 *)
%t A265293 RealDigits[s1 + s2, 10, 120][[1]](* A265293 *)
%Y A265293 Cf. A000129, A002193, A112329, A265291, A265292, A265288 (guide).
%K A265293 nonn,cons
%O A265293 0,1
%A A265293 _Clark Kimberling_, Dec 06 2015