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A190415 Decimal expansion of sum over lower triangular subarray of array G defined at A190404.

%I A190415 #15 Mar 16 2020 09:29:56
%S A190415 1,4,3,6,4,9,6,0,4,3,9,0,2,2,0,4,2,6,0,1,8,5,3,8,1,7,6,0,0,8,5,7,5,4,
%T A190415 5,5,1,0,0,7,0,6,0,0,0,2,8,5,6,2,0,2,4,6,7,3,7,2,4,7,8,9,5,9,6,2,7,6,
%U A190415 5,9,2,9,8,1,4,9,7,0,4,2,2,7,7,1,2,6,9,5,6,2,8,1,8,9,0,4,3,8,8,1,1,2,8,0,7,2,6,7,8,7,0,8
%N A190415 Decimal expansion of sum over lower triangular subarray of array G defined at A190404.
%C A190415 See A190404.
%H A190415 Danny Rorabaugh, <a href="/A190415/b190415.txt">Table of n, a(n) for n = 0..500</a>
%e A190415 0.14364960439022042601853817600857545510070600028562...
%t A190415 f[i_, j_] :=  i + (j + i - 2)(j + i - 1)/2; (* natural number array, A000027 *)
%t A190415 g[i_, j_] := (1/2)^f[i, j];
%t A190415 d[h_] := Sum[g[i,i+h-1], {i,1,250}]; (* h-th up-diag sum *)
%t A190415 e[h_] := Sum[g[i+h,i], {i,1,250}];   (* h-th low-diag sum *)
%t A190415 c1 = N[Sum[d[j], {j, 1, 30}], 50]
%t A190415 RealDigits[c1, 10, 50, -1]  (* A190412 *)
%t A190415 c2 = N[Sum[e[i], {i, 1, 30}], 50]
%t A190415 RealDigits[c2, 10, 50, -1] (* A190415 *)
%t A190415 c1 + c2 (* very close to 1 *)
%o A190415 (Sage)
%o A190415 def A190415(b): # Generate the constant with b bits of precision
%o A190415     return N(sum([sum([(1/2)^(i+j+(j+2*i-2)*(j+2*i-1)/2) for i in range(1,b)]) for j in range(1,b)]),b)
%o A190415 A190415(379) # _Danny Rorabaugh_, Mar 26 2015
%Y A190415 Cf. A190404, A190412.
%K A190415 nonn,cons
%O A190415 0,2
%A A190415 _Clark Kimberling_, May 10 2011
%E A190415 a(50)-a(111) from _Danny Rorabaugh_, Mar 26 2015