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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.
10
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, 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, 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
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
FORMULA
A190404: (1/2)(1 + Sum_{k>=1} (1/2)^T(k)), where T=A000217 (triangular numbers).
A190405: Sum_{k>=1} (1/2)^T(k), where T=A000217.
A190406: Sum_{k>=1} (1/2)^S(k-1), where S=A001844 (centered square numbers).
A190407: Sum_{k>=1} (1/2)^V(k), where V=A058331 (1+2*k^2).
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
nonn,cons,easy
AUTHOR
Clark Kimberling, May 10 2011
STATUS
approved