OFFSET
0,1
COMMENTS
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.
Let G=G(F) denote the array defined by g(i,j)=(1/2)^f(i,j);
R(i)=Sum_{j>=1} g(i,j); i-th row sum of G;
C(j)=Sum_{i>=1} g(i,j); j-th column sum of G;
U(j)=Sum_{i>=1} g(i,i+j-1); j-th upper diagonal sum of G;
L(i)=Sum_{j>=1} g(i+j,j); i-th lower diagonal sum of G;
R(odds)=Sum_{i>=1} R(2i-1); sum, odd numbered rows of G;
R(evens)=Sum_{i>=1} R(2i); sum, even numbered rows of G;
C(odds)=Sum_{j>=1} R(2j-1); sum, odd numbered cols of G;
C(evens)=Sum_{j>=1} R(2j); sum, even numbered cols of G;
UT=Sum_{j>=1} U(j); sum, upper triangular subarray of G;
LT=Sum_{i>=1} L(i); sum, lower triangular subarray of G.
...
Note that R(odds)+R(evens)=C(odds)+C(evens)=UT+LT=1.
...
For the natural number array F=A000027:
R(1)=0.820816280327576933146921385113... (A190404)
R(2)=0.160408140163788466573460692556...
R(3)=0.0177040700818942332867303462782...
R(4)=0.00103953504094711664336517313909...
R(5)=0.0000314862704735583216825865695447...
...
R(odds)=0.838551840434481240061632331355800... (A190408)
R(evens)=0.161448159565518759938367668644199...(A190409)
...
C(1)=0.64163256065515386629... (A190405)
C(2)=0.28326512131030773259...
C(3)=0.066530242620615465175...
C(4)=0.0080604852412309303507...
C(5)=0.00049597048246186070148...
...
C(odds)=0.7086590131172367153696485920526...(A190410)
C(evens)=0.29134098688276328463035140794... (A190411)
...
D(1)=0.53137210011527713548... (A190406)
D(2)=0.25391006493009715683...
D(3)=0.062744200230554270960...
D(4)=0.0078201298601943136650...
D(5)=0.00048840046110854191952...
...
E(1)=0.12695503246504857842... (A190407)
E(2)=0.015686050057638567740...
E(3)=0.00097751623252428920813...
E(4)=0.000030525028819283869970...
E(5)=0.00000047686626214460406264...
...
UT=0.8563503956097795739814618239914245448... (A190412)
LT=0.1436496043902204260185381760085754551... (A190415)
LINKS
Danny Rorabaugh, Table of n, a(n) for n = 0..10000
FORMULA
Equals Product_{k>=1} 1 - 1/(2^(2*k + 1) - 1). - Antonio GraciĆ” Llorente, Oct 01 2024
Equals A299998/2. - Hugo Pfoertner, Oct 01 2024
EXAMPLE
0.820816280327576933146921385113...
MATHEMATICA
f[i_, j_] := i + (j + i - 2)(j + i - 1)/2;
TableForm[Table[f[i, j], {i, 1, 10}, {j, 1, 10}]] (* A000027 *)
r[i_] := Sum[2^-f[i, j], {j, 1, 400}]; (* C(row i) *)
c[j_] := Sum[2^-f[i, j], {i, 1, 400}]; (* C(col j) *)
d[h_] := Sum[2^-f[i, i+h-1], {i, 1, 200}]; (* C(udiag h) *)
e[h_] := Sum[2^-f[i+h, i], {i, 1, 200}]; (* C(ldiag h) *)
RealDigits[r[1], 10, 120, -1] (* A190404 *)
N[r[1], 30]
N[r[2], 30]
N[r[3], 30]
N[r[4], 30]
N[r[5], 30]
N[r[6], 30]
RealDigits[c[1], 10, 120, -1] (* A190405 *)
N[c[1], 20]
N[c[2], 20]
N[c[3], 20]
N[c[4], 20]
N[c[5], 20]
N[c[6], 20]
RealDigits[d[1], 10, 20, -1] (* A190406 *)
N[d[1], 20]
N[d[2], 20]
N[d[3], 20]
N[d[4], 20]
N[d[5], 20]
N[d[6], 20]
RealDigits[e[1], 10, 20, -1] (* A190407 *)
N[e[1], 20]
N[e[2], 20]
N[e[3], 20]
N[e[4], 20]
N[e[5], 20]
N[e[6], 20]
PROG
(Sage)
def A190404(b): # Generate the constant with b bits of precision
return N(sum([(1/2)^(1+j*(j+1)/2) for j in range(1, b)])+1/2, b)
A190404(409) # Danny Rorabaugh, Mar 25 2015
CROSSREFS
KEYWORD
AUTHOR
Clark Kimberling, May 10 2011
STATUS
approved