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.

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

Author

Clark Kimberling, May 10 2011

Keywords

• nonn
• cons
• easy

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)

Examples

			0.820816280327576933146921385113...
		

Links

• Danny Rorabaugh, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A190405-A190412, A190415, A299998.

Programs

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]

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

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 Graciá Llorente, Oct 01 2024

Equals A299998/2. - Hugo Pfoertner, Oct 01 2024