A190406 -id:A190406 - cp's OEIS frontend

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

Clark Kimberling, May 10 2011

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)

			0.820816280327576933146921385113...

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

Cf. A190405-A190412, A190415, A299998.

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]

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

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

A190405 Decimal expansion of Sum_{k>=1} (1/2)^T(k), where T=A000217 (triangular numbers); based on column 1 of the natural number array, A000027.

Original entry on oeis.org

6, 4, 1, 6, 3, 2, 5, 6, 0, 6, 5, 5, 1, 5, 3, 8, 6, 6, 2, 9, 3, 8, 4, 2, 7, 7, 0, 2, 2, 5, 4, 2, 9, 4, 3, 4, 2, 2, 6, 0, 6, 1, 5, 3, 7, 9, 5, 6, 7, 3, 9, 7, 4, 7, 8, 0, 4, 6, 5, 1, 6, 2, 2, 3, 8, 0, 1, 4, 4, 6, 0, 3, 7, 3, 3, 3, 5, 1, 7, 7, 5, 6, 0, 0, 3, 6, 4, 1, 7, 1, 6, 2, 3, 3, 5, 9, 1, 3, 3, 0, 8, 6, 0, 8, 9, 7, 3, 5, 3, 1, 6, 3, 4, 3, 6, 1, 9, 4, 6, 1

Offset: 0

Clark Kimberling, May 10 2011

See A190404.
Binary expansion is .1010010001... (A023531). - Rick L. Shepherd, Jan 05 2014
From Amiram Eldar, Dec 07 2020: (Start)
This constant is not a quadratic irrational (Duverney, 1995).
The Engel expansion of this constant are the powers of 2 (A000079) above 1. (End)

			0.64163256065515386629...

Danny Rorabaugh, Table of n, a(n) for n = 0..10000
Daniel Duverney, Sommes de deux carrés et irrationalité de valeurs de fonctions thêta, Comptes rendus de l'Académie des sciences, Série 1, Mathématique, Vol. 320, No. 9 (1995), pp. 1041-1044.

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 (triangular numbers).
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).
Cf. A000079.

RealDigits[EllipticTheta[2, 0, 1/Sqrt[2]]/2^(7/8) - 1, 10, 120] // First (* Jean-François Alcover, Feb 12 2013 *)
RealDigits[Total[(1/2)^Accumulate[Range[50]]],10,120][[1]] (* Harvey P. Dale, Oct 18 2013 *)
(* See also A190404 *)

th2(x)=2*x^.25 + 2*suminf(n=1,x^(n+1/2)^2)
th2(sqrt(.5))/2^(7/8)-1 \\ Charles R Greathouse IV, Jun 06 2016

def A190405(b):  # Generate the constant with b bits of precision
    return N(sum([(1/2)^(j*(j+1)/2) for j in range(1,b)]),b)
A190405(409) # Danny Rorabaugh, Mar 25 2015

A190407 Decimal expansion of Sum_{k>=1} (1/2)^A058331(k); based on a diagonal of the natural number array, A000027.

Original entry on oeis.org

1, 2, 6, 9, 5, 5, 0, 3, 2, 4, 6, 5, 0, 4, 8, 5, 7, 8, 4, 1, 6, 2, 5, 0, 5, 4, 3, 6, 3, 5, 7, 2, 5, 6, 7, 8, 8, 0, 6, 9, 6, 2, 1, 6, 8, 1, 9, 0, 1, 4, 6, 8, 0, 0, 2, 3, 1, 5, 0, 6, 1, 7, 8, 4, 9, 2, 5, 0, 9, 9, 2, 2, 7, 6, 2, 2, 7, 3, 0, 7, 5, 3, 8, 2, 1, 6, 5, 1, 3, 8, 3, 2, 3, 0, 9, 6, 1, 4, 3, 1, 3, 9, 1, 4, 3, 1, 4, 5, 8, 3, 2, 5, 4, 2, 1, 3, 0, 3, 3, 2

Offset: 0

Clark Kimberling, May 10 2011

See A190404.

			0.12695503246504857842...

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

Cf. A058331, A190404, A190405, A190406.

(* See also A190404 *)
RealDigits[(EllipticTheta[3, 0, 1/4]-1)/4, 10, 120] // First (* Jean-François Alcover, Feb 13 2013 *)

th3(x)=1 + 2*suminf(n=1,x^n^2)
(th3(1/4)-1)/4 \\ Charles R Greathouse IV, Jun 06 2016

def A190407(b): # Generate the constant with b bits of precision
    return N(sum([(1/2)^(2*k^2+1) for k in range(1,b)]),b)
A190407(415) # Danny Rorabaugh, Mar 26 2015

Equals Sum_{k>=1} (1/2)^V(k), where V=A058331 (1+2*k^2).

cp's OEIS Frontend

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.

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A190405 Decimal expansion of Sum_{k>=1} (1/2)^T(k), where T=A000217 (triangular numbers); based on column 1 of the natural number array, A000027.

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A190407 Decimal expansion of Sum_{k>=1} (1/2)^A058331(k); based on a diagonal of the natural number array, A000027.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Sage

Formula