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A265296 Decimal expansion of Sum_{n >= 1} (c(2*n) - c(2*n-1)), where c(n) = the n-th convergent to x = sqrt(3).

%I A265296 #13 Aug 30 2022 14:12:50
%S A265296 1,0,8,9,8,6,3,1,7,0,8,8,7,0,0,3,2,2,7,8,8,9,3,2,5,7,2,1,1,3,9,7,2,5,
%T A265296 8,1,2,8,8,2,5,1,4,1,9,7,7,5,9,6,9,9,9,6,4,9,5,6,4,5,8,6,7,8,2,9,8,0,
%U A265296 2,4,4,7,2,5,5,5,8,6,8,3,0,8,6,2,6,2
%N A265296 Decimal expansion of Sum_{n >= 1} (c(2*n) - c(2*n-1)), where c(n) = the n-th convergent to x = sqrt(3).
%F A265296 Equals 2*sqrt(3)*Sum_{n >= 1} x^(n^2)*(1 + x^(2*n))/(1 - x^(2*n)), where x = 2 - sqrt(3). - _Peter Bala_, Aug 24 2022
%e A265296 sum = 1.0898631708870032278893257211397258128825141977596999...
%p A265296 x := 2 - sqrt(3):
%p A265296 evalf(2*sqrt(3)*add(x^(n^2)*(1 + x^(2*n))/(1 - x^(2*n)), n = 1..13), 100); # _Peter Bala_, Aug 24 2022
%t A265296 x = Sqrt[3]; z = 600; c = Convergents[x, z];
%t A265296 s1 = Sum[x - c[[2 k - 1]], {k, 1, z/2}]; N[s1, 200]
%t A265296 s2 = Sum[c[[2 k]] - x, {k, 1, z/2}]; N[s2, 200]
%t A265296 N[s1 + s2, 200]
%t A265296 RealDigits[s1, 10, 120][[1]]  (* A265294 *)
%t A265296 RealDigits[s2, 10, 120][[1]]  (* A265295 *)
%t A265296 RealDigits[s1 + s2, 10, 120][[1]](* A265296 *)
%Y A265296 Cf. A002194, A002530, A002531, A265294, A265295, A265288 (guide).
%K A265296 nonn,cons
%O A265296 1,3
%A A265296 _Clark Kimberling_, Dec 07 2015