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A190404 Decimal expansion of (1/2)(1 + Sum_{k>=1} (1/2)^T(k)), where T=A000217 (triangular numbers); based on row 1 of the natural number array, A000027.

%I A190404 #32 Oct 01 2024 13:17:01
%S A190404 8,2,0,8,1,6,2,8,0,3,2,7,5,7,6,9,3,3,1,4,6,9,2,1,3,8,5,1,1,2,7,1,4,7,
%T A190404 1,7,1,1,3,0,3,0,7,6,8,9,7,8,3,6,9,8,7,3,9,0,2,3,2,5,8,1,1,1,9,0,0,7,
%U A190404 2,3,0,1,8,6,6,6,7,5,8,8,7,8,0,0,1,8,2,0,8,5,8,1,1,6,7,9,5,6,6,5,4,3,0,4,4,8,6,7,6,5,8,1,7,1,8,0,9,7,3,0
%N A190404 Decimal expansion of (1/2)(1 + Sum_{k>=1} (1/2)^T(k)), where T=A000217 (triangular numbers); based on row 1 of the natural number array, A000027.
%C A190404 Suppose that F={f(i,j): i>=1, j>=1} is an array of positive integers such that every positive integer occurs exactly once in F.
%C A190404 Let G=G(F) denote the array defined by g(i,j)=(1/2)^f(i,j);
%C A190404 R(i)=Sum_{j>=1} g(i,j); i-th row sum of G;
%C A190404 C(j)=Sum_{i>=1} g(i,j); j-th column sum of G;
%C A190404 U(j)=Sum_{i>=1} g(i,i+j-1); j-th upper diagonal sum of G;
%C A190404 L(i)=Sum_{j>=1} g(i+j,j); i-th lower diagonal sum of G;
%C A190404 R(odds)=Sum_{i>=1} R(2i-1); sum, odd numbered rows of G;
%C A190404 R(evens)=Sum_{i>=1} R(2i); sum, even numbered rows of G;
%C A190404 C(odds)=Sum_{j>=1} R(2j-1); sum, odd numbered cols of G;
%C A190404 C(evens)=Sum_{j>=1} R(2j); sum, even numbered cols of G;
%C A190404 UT=Sum_{j>=1} U(j); sum, upper triangular subarray of G;
%C A190404 LT=Sum_{i>=1} L(i); sum, lower triangular subarray of G.
%C A190404 ...
%C A190404 Note that R(odds)+R(evens)=C(odds)+C(evens)=UT+LT=1.
%C A190404 ...
%C A190404 For the natural number array F=A000027:
%C A190404 R(1)=0.820816280327576933146921385113... (A190404)
%C A190404 R(2)=0.160408140163788466573460692556...
%C A190404 R(3)=0.0177040700818942332867303462782...
%C A190404 R(4)=0.00103953504094711664336517313909...
%C A190404 R(5)=0.0000314862704735583216825865695447...
%C A190404 ...
%C A190404 R(odds)=0.838551840434481240061632331355800... (A190408)
%C A190404 R(evens)=0.161448159565518759938367668644199...(A190409)
%C A190404 ...
%C A190404 C(1)=0.64163256065515386629... (A190405)
%C A190404 C(2)=0.28326512131030773259...
%C A190404 C(3)=0.066530242620615465175...
%C A190404 C(4)=0.0080604852412309303507...
%C A190404 C(5)=0.00049597048246186070148...
%C A190404 ...
%C A190404 C(odds)=0.7086590131172367153696485920526...(A190410)
%C A190404 C(evens)=0.29134098688276328463035140794... (A190411)
%C A190404 ...
%C A190404 D(1)=0.53137210011527713548... (A190406)
%C A190404 D(2)=0.25391006493009715683...
%C A190404 D(3)=0.062744200230554270960...
%C A190404 D(4)=0.0078201298601943136650...
%C A190404 D(5)=0.00048840046110854191952...
%C A190404 ...
%C A190404 E(1)=0.12695503246504857842... (A190407)
%C A190404 E(2)=0.015686050057638567740...
%C A190404 E(3)=0.00097751623252428920813...
%C A190404 E(4)=0.000030525028819283869970...
%C A190404 E(5)=0.00000047686626214460406264...
%C A190404 ...
%C A190404 UT=0.8563503956097795739814618239914245448... (A190412)
%C A190404 LT=0.1436496043902204260185381760085754551... (A190415)
%H A190404 Danny Rorabaugh, <a href="/A190404/b190404.txt">Table of n, a(n) for n = 0..10000</a>
%F A190404 A190404: (1/2)(1 + Sum_{k>=1} (1/2)^T(k)), where T=A000217 (triangular numbers).
%F A190404 A190405: Sum_{k>=1} (1/2)^T(k), where T=A000217.
%F A190404 A190406: Sum_{k>=1} (1/2)^S(k-1), where S=A001844 (centered square numbers).
%F A190404 A190407: Sum_{k>=1} (1/2)^V(k), where V=A058331 (1+2*k^2).
%F A190404 Equals Product_{k>=1} 1 - 1/(2^(2*k + 1) - 1). - _Antonio Graciá Llorente_, Oct 01 2024
%F A190404 Equals A299998/2. - _Hugo Pfoertner_, Oct 01 2024
%e A190404 0.820816280327576933146921385113...
%t A190404 f[i_, j_] := i + (j + i - 2)(j + i - 1)/2;
%t A190404 TableForm[Table[f[i,j],{i,1,10},{j,1,10}]] (* A000027 *)
%t A190404 r[i_] := Sum[2^-f[i, j], {j,1,400}];    (* C(row i) *)
%t A190404 c[j_] := Sum[2^-f[i,j], {i,1,400}];     (* C(col j) *)
%t A190404 d[h_] := Sum[2^-f[i,i+h-1], {i,1,200}]; (* C(udiag h) *)
%t A190404 e[h_] := Sum[2^-f[i+h,i], {i,1,200}];   (* C(ldiag h) *)
%t A190404 RealDigits[r[1], 10, 120, -1]  (* A190404 *)
%t A190404 N[r[1], 30]
%t A190404 N[r[2], 30]
%t A190404 N[r[3], 30]
%t A190404 N[r[4], 30]
%t A190404 N[r[5], 30]
%t A190404 N[r[6], 30]
%t A190404 RealDigits[c[1], 10, 120, -1] (* A190405 *)
%t A190404 N[c[1], 20]
%t A190404 N[c[2], 20]
%t A190404 N[c[3], 20]
%t A190404 N[c[4], 20]
%t A190404 N[c[5], 20]
%t A190404 N[c[6], 20]
%t A190404 RealDigits[d[1], 10, 20, -1] (* A190406 *)
%t A190404 N[d[1], 20]
%t A190404 N[d[2], 20]
%t A190404 N[d[3], 20]
%t A190404 N[d[4], 20]
%t A190404 N[d[5], 20]
%t A190404 N[d[6], 20]
%t A190404 RealDigits[e[1], 10, 20, -1] (* A190407 *)
%t A190404 N[e[1], 20]
%t A190404 N[e[2], 20]
%t A190404 N[e[3], 20]
%t A190404 N[e[4], 20]
%t A190404 N[e[5], 20]
%t A190404 N[e[6], 20]
%o A190404 (Sage)
%o A190404 def A190404(b):  # Generate the constant with b bits of precision
%o A190404     return N(sum([(1/2)^(1+j*(j+1)/2) for j in range(1,b)])+1/2,b)
%o A190404 A190404(409) # _Danny Rorabaugh_, Mar 25 2015
%Y A190404 Cf. A190405-A190412, A190415, A299998.
%K A190404 nonn,cons,easy
%O A190404 0,1
%A A190404 _Clark Kimberling_, May 10 2011