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A265295 Decimal expansion of Sum_{n >= 1} (c(2*n) - x), where c(n) = the n-th convergent to x = sqrt(3).

%I A265295 #12 Aug 30 2022 14:12:36
%S A265295 2,8,7,2,8,0,0,8,0,0,8,3,4,8,8,3,9,3,5,1,1,4,5,1,5,3,9,8,7,6,6,8,3,3,
%T A265295 1,6,8,2,3,9,0,9,4,2,0,8,6,4,5,6,7,1,8,7,9,3,8,7,1,6,8,2,6,8,1,3,8,8,
%U A265295 3,8,6,4,1,0,7,1,6,8,0,0,6,4,0,8,2,6
%N A265295 Decimal expansion of Sum_{n >= 1} (c(2*n) - x), where c(n) = the n-th convergent to x = sqrt(3).
%F A265295 Equals 2*sqrt(3)*Sum_{n >= 1} x^(n^2)*(1 + x^n)/(1 - x^n), where x = 7 - 4*sqrt(3). - _Peter Bala_, Aug 24 2022
%e A265295 sum = 0.28728008008348839351145153987668331682390...
%p A265295 x := 7 - 4*sqrt(3):
%p A265295 evalf(2*sqrt(3)*add( x^(n^2)*(1 + x^n)/(1 - x^n), n = 1..10), 100); # _Peter Bala_, Aug 24 2022
%t A265295 x = Sqrt[3]; z = 600; c = Convergents[x, z];
%t A265295 s1 = Sum[x - c[[2 k - 1]], {k, 1, z/2}]; N[s1, 200]
%t A265295 s2 = Sum[c[[2 k]] - x, {k, 1, z/2}]; N[s2, 200]
%t A265295 N[s1 + s2, 200]
%t A265295 RealDigits[s1, 10, 120][[1]]  (* A265294 *)
%t A265295 RealDigits[s2, 10, 120][[1]]  (* A265295 *)
%t A265295 RealDigits[s1 + s2, 10, 120][[1]](* A265296 *)
%Y A265295 Cf. A002194, A002530, A002531, A265294, A265296, A265288 (guide).
%K A265295 nonn,cons
%O A265295 0,1
%A A265295 _Clark Kimberling_, Dec 07 2015