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A190412 Decimal expansion of sum over upper triangular subarray of array G defined at A190404.

%I A190412 #18 Mar 16 2020 09:29:52
%S A190412 8,5,6,3,5,0,3,9,5,6,0,9,7,7,9,5,7,3,9,8,1,4,6,1,8,2,3,9,9,1,4,2,4,5,
%T A190412 4,4,8,9,9,2,9,3,9,9,9,7,1,4,3,7,9,7,5,3,2,6,2,7,5,2,1,0,4,0,3,7,2,3,
%U A190412 4,0,7,0,1,8,5,0,2,9,5,7,7,2,2,8,7,3,0,4,3,7,1,8,1,0,9,5,6,1,1,8,8,7,1,9,2,7
%N A190412 Decimal expansion of sum over upper triangular subarray of array G defined at A190404.
%C A190412 See A190404.
%H A190412 Danny Rorabaugh, <a href="/A190412/b190412.txt">Table of n, a(n) for n = 0..500</a>
%e A190412 0.85635039560977957398146182399142454489929399971437975...
%t A190412 f[i_, j_] :=  i + (j + i - 2)(j + i - 1)/2; (* natural number array, A000027 *)
%t A190412 g[i_, j_] := (1/2)^f[i, j];
%t A190412 d[h_] := Sum[g[i,i+h-1], {i,1,250}]; (* h-th up-diag sum *)
%t A190412 e[h_] := Sum[g[i+h,i], {i,1,250}];   (* h-th low-diag sum *)
%t A190412 c1 = N[Sum[d[j], {j, 1, 30}], 50]
%t A190412 RealDigits[c1, 10, 50, -1]  (* A190412 *)
%t A190412 c2 = N[Sum[e[i], {i, 1, 30}], 50]
%t A190412 RealDigits[c2, 10, 50, -1] (* A190415 *)
%t A190412 c1 + c2 (* very close to 1 *)
%o A190412 (Sage)
%o A190412 def A190412(b): # Generate the constant with b bits of precision
%o A190412     return N(sum([sum([(1/2)^(i+(j+2*i-3)*(j+2*i-2)/2) for i in range(1,b)]) for j in range(1,b)]),b)
%o A190412 A190412(365) # _Danny Rorabaugh_, Mar 26 2015
%Y A190412 Cf. A190404, A190415.
%K A190412 nonn,cons
%O A190412 0,1
%A A190412 _Clark Kimberling_, May 10 2011
%E A190412 a(49) corrected and a(50)-a(105) added by _Danny Rorabaugh_, Mar 24 2015