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

0,1

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

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

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

Cf. A190405-A190412, A190415, A299998.

Sequence in context: A182170 A011105 A098829 * A243433 A080729 A262080

Adjacent sequences: A190401 A190402 A190403 * A190405 A190406 A190407

Keyword

nonn,cons,easy

Author

Clark Kimberling, May 10 2011

Status

approved