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A265294 Decimal expansion of Sum_{n>=1} (x - c(2n-1)), where c = convergents to (x = sqrt(3)).

%I A265294 #13 Aug 30 2022 14:12:27
%S A265294 8,0,2,5,8,3,0,9,0,8,0,3,5,1,4,8,3,4,3,7,7,8,7,4,1,8,1,2,6,3,0,4,2,4,
%T A265294 9,6,0,5,8,6,0,4,7,7,6,8,9,5,1,3,2,7,7,7,0,1,7,7,4,1,8,5,1,4,8,4,1,4,
%U A265294 0,6,0,8,4,4,8,7,0,0,3,0,2,2,1,7,9,4
%N A265294 Decimal expansion of Sum_{n>=1} (x - c(2n-1)), where c = convergents to (x = sqrt(3)).
%F A265294 From _Peter Bala_, Aug 24 2022: (Start)
%F A265294 Equals 2*sqrt(3)*Sum_{n >= 1} 1/( 1 + (2+sqrt(3))^(2*n-1) ).
%F A265294 A more rapidly converging series for the constant is 2*sqrt(3)*Sum_{n >= 1} x^(n*(n+1)/2)/(x^n - 1), where x = sqrt(3) - 2. See A001227. (End)
%e A265294 sum = 0.8025830908035148343778741812630...
%p A265294 x := sqrt(3) - 2:
%p A265294 evalf(2*sqrt(3)*add( x^(n*(n+1)/2)/(x^n - 1), n = 1..18), 100); # Peter Bala, Aug 24 2022
%t A265294 x = Sqrt[3]; z = 600; c = Convergents[x, z];
%t A265294 s1 = Sum[x - c[[2 k - 1]], {k, 1, z/2}]; N[s1, 200]
%t A265294 s2 = Sum[c[[2 k]] - x, {k, 1, z/2}]; N[s2, 200]
%t A265294 N[s1 + s2, 200]
%t A265294 RealDigits[s1, 10, 120][[1]]  (* A265294 *)
%t A265294 RealDigits[s2, 10, 120][[1]]  (* A265295 *)
%t A265294 RealDigits[s1 + s2, 10, 120][[1]](* A265296 *)
%Y A265294 Cf. A002194, A002530, A002531, A265295, A265296, A265288 (guide).
%K A265294 nonn,cons
%O A265294 0,1
%A A265294 _Clark Kimberling_, Dec 07 2015