A190404 - OEIS

%I #32 Oct 01 2024 13:17:01

%S 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 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 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 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 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 Let G=G(F) denote the array defined by g(i,j)=(1/2)^f(i,j);

%C R(i)=Sum_{j>=1} g(i,j); i-th row sum of G;

%C C(j)=Sum_{i>=1} g(i,j); j-th column sum of G;

%C U(j)=Sum_{i>=1} g(i,i+j-1); j-th upper diagonal sum of G;

%C L(i)=Sum_{j>=1} g(i+j,j); i-th lower diagonal sum of G;

%C R(odds)=Sum_{i>=1} R(2i-1); sum, odd numbered rows of G;

%C R(evens)=Sum_{i>=1} R(2i); sum, even numbered rows of G;

%C C(odds)=Sum_{j>=1} R(2j-1); sum, odd numbered cols of G;

%C C(evens)=Sum_{j>=1} R(2j); sum, even numbered cols of G;

%C UT=Sum_{j>=1} U(j); sum, upper triangular subarray of G;

%C LT=Sum_{i>=1} L(i); sum, lower triangular subarray of G.

%C ...

%C Note that R(odds)+R(evens)=C(odds)+C(evens)=UT+LT=1.

%C ...

%C For the natural number array F=A000027:

%C R(1)=0.820816280327576933146921385113... (A190404)

%C R(2)=0.160408140163788466573460692556...

%C R(3)=0.0177040700818942332867303462782...

%C R(4)=0.00103953504094711664336517313909...

%C R(5)=0.0000314862704735583216825865695447...

%C ...

%C R(odds)=0.838551840434481240061632331355800... (A190408)

%C R(evens)=0.161448159565518759938367668644199...(A190409)

%C ...

%C C(1)=0.64163256065515386629... (A190405)

%C C(2)=0.28326512131030773259...

%C C(3)=0.066530242620615465175...

%C C(4)=0.0080604852412309303507...

%C C(5)=0.00049597048246186070148...

%C ...

%C C(odds)=0.7086590131172367153696485920526...(A190410)

%C C(evens)=0.29134098688276328463035140794... (A190411)

%C ...

%C D(1)=0.53137210011527713548... (A190406)

%C D(2)=0.25391006493009715683...

%C D(3)=0.062744200230554270960...

%C D(4)=0.0078201298601943136650...

%C D(5)=0.00048840046110854191952...

%C ...

%C E(1)=0.12695503246504857842... (A190407)

%C E(2)=0.015686050057638567740...

%C E(3)=0.00097751623252428920813...

%C E(4)=0.000030525028819283869970...

%C E(5)=0.00000047686626214460406264...

%C ...

%C UT=0.8563503956097795739814618239914245448... (A190412)

%C LT=0.1436496043902204260185381760085754551... (A190415)

%H Danny Rorabaugh, <a href="/A190404/b190404.txt">Table of n, a(n) for n = 0..10000</a>

%F A190404: (1/2)(1 + Sum_{k>=1} (1/2)^T(k)), where T=A000217 (triangular numbers).

%F A190405: Sum_{k>=1} (1/2)^T(k), where T=A000217.

%F A190406: Sum_{k>=1} (1/2)^S(k-1), where S=A001844 (centered square numbers).

%F A190407: Sum_{k>=1} (1/2)^V(k), where V=A058331 (1+2*k^2).

%F Equals Product_{k>=1} 1 - 1/(2^(2*k + 1) - 1). - _Antonio Graciá Llorente_, Oct 01 2024

%F Equals A299998/2. - _Hugo Pfoertner_, Oct 01 2024

%e 0.820816280327576933146921385113...

%t f[i_, j_] := i + (j + i - 2)(j + i - 1)/2;

%t TableForm[Table[f[i,j],{i,1,10},{j,1,10}]] (* A000027 *)

%t r[i_] := Sum[2^-f[i, j], {j,1,400}];    (* C(row i) *)

%t c[j_] := Sum[2^-f[i,j], {i,1,400}];     (* C(col j) *)

%t d[h_] := Sum[2^-f[i,i+h-1], {i,1,200}]; (* C(udiag h) *)

%t e[h_] := Sum[2^-f[i+h,i], {i,1,200}];   (* C(ldiag h) *)

%t RealDigits[r[1], 10, 120, -1]  (* A190404 *)

%t N[r[1], 30]

%t N[r[2], 30]

%t N[r[3], 30]

%t N[r[4], 30]

%t N[r[5], 30]

%t N[r[6], 30]

%t RealDigits[c[1], 10, 120, -1] (* A190405 *)

%t N[c[1], 20]

%t N[c[2], 20]

%t N[c[3], 20]

%t N[c[4], 20]

%t N[c[5], 20]

%t N[c[6], 20]

%t RealDigits[d[1], 10, 20, -1] (* A190406 *)

%t N[d[1], 20]

%t N[d[2], 20]

%t N[d[3], 20]

%t N[d[4], 20]

%t N[d[5], 20]

%t N[d[6], 20]

%t RealDigits[e[1], 10, 20, -1] (* A190407 *)

%t N[e[1], 20]

%t N[e[2], 20]

%t N[e[3], 20]

%t N[e[4], 20]

%t N[e[5], 20]

%t N[e[6], 20]

%o (Sage)

%o def A190404(b):  # Generate the constant with b bits of precision

%o     return N(sum([(1/2)^(1+j*(j+1)/2) for j in range(1,b)])+1/2,b)

%o A190404(409) # _Danny Rorabaugh_, Mar 25 2015

%Y Cf. A190405-A190412, A190415, A299998.

%K nonn,cons,easy

%O 0,1

%A _Clark Kimberling_, May 10 2011