cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-9 of 9 results.

A190408 Decimal expansion of sum of odd-numbered rows of array G defined at A190404.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, May 10 2011

Keywords

Comments

See A190404.

Examples

			0.8385518404344812400616323313558001448737569046651...

Links

Crossrefs

Cf. A190404, A190409.

Programs

  • Mathematica
    f[i_, j_] :=  i + (j + i - 2)(j + i - 1)/2; (* natural number array, A000027 *)
g[i_, j_] := (1/2)^f[i, j]; (* array G *)
r[i_] := Sum[g[i,j], {j,1,400}];  (* i-th row sum of G *)
c1 = N[Sum[r[2 i - 1], {i, 1, 10}], 60]
RealDigits[c1, 10, 60, -1] (* A190408 *)
c2 = N[Sum[r[2 i], {i, 1, 10}], 60]
RealDigits[c2, 10, 60, -1] (* A190409 *)
c1 + c2 (* very close to 1 *)
  • Sage
    def A190408(b): # Generate the constant with b bits of precision
    return N(sum([sum([(1/2)^(i+(j+i-2)*(j+i-1)/2) for j in range(1,b)]) for i in range(1,b,2)]),b)
A190408(405) # Danny Rorabaugh, Mar 26 2015

Extensions

a(69)-a(119) corrected by Danny Rorabaugh, Mar 26 2015

A190409 Decimal expansion of sum of even-numbered rows of array G defined at A190404.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, May 10 2011

Keywords

Comments

See A190404.

Examples

			0.161448159565518759938367668644199855126243095334825...

Links

Crossrefs

Cf. A190404, A190408.

Programs

  • Mathematica
    f[i_, j_] :=  i + (j + i - 2) (j + i - 1)/2; (* natural number array, A000027 *)
g[i_,j_] := (1/2)^f[i, j]; (* array G *)
r[i_] := Sum[g[i, j], {j,1,400}]; (* i-th row sum of G *)
c1 = N[Sum[r[2 i - 1], {i, 1, 10}], 60]
RealDigits[c1, 10, 60, -1] (* A190408 *)
c2 = N[Sum[r[2 i], {i, 1, 10}], 60]
RealDigits[c2, 10, 60, -1] (* A190409 *)
c1 + c2 (* very close to 1 *)
  • Sage
    def A190409(b): # Generate the constant with b bits of precision
    return N(sum([sum([(1/2)^(i+(j+i-2)*(j+i-1)/2) for j in range(1,b)]) for i in range(2,b,2)]),b)
A190409(350) # Danny Rorabaugh, Mar 25 2015

Extensions

a(75)-a(103) corrected by Danny Rorabaugh, Mar 24 2015

A190410 Decimal expansion of sum of odd-numbered columns of array G defined at A190404.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, May 10 2011

Keywords

Comments

See A190404.

Examples

			0.708659013117236715369648592052676333201596941320...

Links

Crossrefs

Cf. A190404, A190411.

Programs

  • Mathematica
    f[i_, j_] :=  i + (j + i - 2) (j + i - 1)/2; (* natural number array, A000027 *)
g[i_, j_] := (1/2)^f[i, j];
c[j_] := Sum[g[i,j], {i,1,400}]; (* j-th column sum of G *)
c1 = N[Sum[c[2 i - 1], {i, 1, 10}], 60]
RealDigits[c1, 10, 60, -1]  (* A190410 *)
c2 = N[Sum[c[2 i], {i, 1, 10}], 60]
RealDigits[c2, 10, 60, -1]  (* A190411 *)
c1 + c2 (* very close to 1 *)
  • Sage
    def A190410(b): # Generate the constant with b bits of precision
    return N(sum([sum([(1/2)^(i+(j+i-2)*(j+i-1)/2) for i in range(1,b)]) for j in range(1,b,2)]),b)
A190410(350) # Danny Rorabaugh, Mar 24 2015

Extensions

a(62)-a(79) corrected and a(80)-a(103) added by Danny Rorabaugh, Mar 24 2015

A190411 Decimal expansion of sum of even-numbered columns of array G defined at A190404.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, May 10 2011

Keywords

Comments

See A190404.

Examples

			0.29134098688276328463035140794732366679840305867959490...

Links

Crossrefs

Cf. A190404, A190410.

Programs

  • Mathematica
    f[i_, j_] :=  i + (j + i - 2)(j + i - 1)/2; (* natural number array, A000027 *)
g[i_, j_] := (1/2)^f[i, j];
c[j_] := Sum[g[i,j], {i,1,400}]; (* j-th column sum of G *)
c1 = N[Sum[c[2 i - 1], {i, 1, 10}], 60]
RealDigits[c1, 10, 60, -1]  (* A190410 *)
c2 = N[Sum[c[2 i], {i, 1, 10}], 60]
RealDigits[c2, 10, 60, -1]  (* A190411 *)
c1 + c2 (* very close to 1 *)

Extensions

a(69)-a(79) corrected and a(80)-a(115) added by Danny Rorabaugh, Mar 26 2015

A190412 Decimal expansion of sum over upper triangular subarray of array G defined at A190404.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, May 10 2011

Keywords

Comments

See A190404.

Examples

			0.85635039560977957398146182399142454489929399971437975...

Links

Crossrefs

Cf. A190404, A190415.

Programs

  • Mathematica
    f[i_, j_] :=  i + (j + i - 2)(j + i - 1)/2; (* natural number array, A000027 *)
g[i_, j_] := (1/2)^f[i, j];
d[h_] := Sum[g[i,i+h-1], {i,1,250}]; (* h-th up-diag sum *)
e[h_] := Sum[g[i+h,i], {i,1,250}];   (* h-th low-diag sum *)
c1 = N[Sum[d[j], {j, 1, 30}], 50]
RealDigits[c1, 10, 50, -1]  (* A190412 *)
c2 = N[Sum[e[i], {i, 1, 30}], 50]
RealDigits[c2, 10, 50, -1] (* A190415 *)
c1 + c2 (* very close to 1 *)
  • Sage
    def A190412(b): # Generate the constant with b bits of precision
    return N(sum([sum([(1/2)^(i+(j+2*i-3)*(j+2*i-2)/2) for i in range(1,b)]) for j in range(1,b)]),b)
A190412(365) # Danny Rorabaugh, Mar 26 2015

Extensions

a(49) corrected and a(50)-a(105) added by Danny Rorabaugh, Mar 24 2015

A190415 Decimal expansion of sum over lower triangular subarray of array G defined at A190404.

Original entry on oeis.org

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

Views

Author

Clark Kimberling, May 10 2011

Keywords

Comments

See A190404.

Examples

			0.14364960439022042601853817600857545510070600028562...

Links

Crossrefs

Cf. A190404, A190412.

Programs

  • Mathematica
    f[i_, j_] :=  i + (j + i - 2)(j + i - 1)/2; (* natural number array, A000027 *)
g[i_, j_] := (1/2)^f[i, j];
d[h_] := Sum[g[i,i+h-1], {i,1,250}]; (* h-th up-diag sum *)
e[h_] := Sum[g[i+h,i], {i,1,250}];   (* h-th low-diag sum *)
c1 = N[Sum[d[j], {j, 1, 30}], 50]
RealDigits[c1, 10, 50, -1]  (* A190412 *)
c2 = N[Sum[e[i], {i, 1, 30}], 50]
RealDigits[c2, 10, 50, -1] (* A190415 *)
c1 + c2 (* very close to 1 *)
  • Sage
    def A190415(b): # Generate the constant with b bits of precision
    return N(sum([sum([(1/2)^(i+j+(j+2*i-2)*(j+2*i-1)/2) for i in range(1,b)]) for j in range(1,b)]),b)
A190415(379) # Danny Rorabaugh, Mar 26 2015

Extensions

a(50)-a(111) from Danny Rorabaugh, Mar 26 2015

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

Views

Author

Clark Kimberling, May 10 2011

Keywords

Comments

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)

Examples

			0.64163256065515386629...

Links

Crossrefs

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.

Programs

  • Mathematica
    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 *)
  • PARI
    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
  • Sage
    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

A190406 Decimal expansion of Sum_{k>=1} (1/2)^S(k-1), where S=A001844 (centered square numbers).

Original entry on oeis.org

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

Views

Author

Clark Kimberling, May 10 2011

Keywords

Comments

See A190404.

Links

Crossrefs

Cf. A190404-A190407.

Programs

  • Maple
    evalf(JacobiTheta2(0,1/4)/2^(3/2)) ; # R. J. Mathar, Jul 15 2013
  • Mathematica
    (See A190404.)
(* or *) RealDigits[EllipticTheta[2, 0, 1/4]/(2*Sqrt[2]), 10, 120] // First (* Jean-François Alcover, Feb 12 2013 *)
  • PARI
    th2(x)=x^.25 + 2*suminf(n=1,x^(n+1/2)^2)
th2(1/4)/sqrt(8) \\ Charles R Greathouse IV, Jun 06 2016
  • Sage
    def A190406(b): # Generate the constant with b bits of precision
    return N(sum([(1/2)^(2*j*(j+1)+1) for j in range(0,b)]),b)
A190406(409) # Danny Rorabaugh, Mar 26 2015

Formula

a(n) = floor(10^(n+1)*Sum_{j>=0} (1/2)^(2*j*(j+1)+1)) mod 10. - Danny Rorabaugh, Mar 26 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

Views

Author

Clark Kimberling, May 10 2011

Keywords

Comments

See A190404.

Examples

			0.12695503246504857842...

Links

Crossrefs

Cf. A058331, A190404, A190405, A190406.

Programs

Formula

Equals Sum_{k>=1} (1/2)^V(k), where V=A058331 (1+2*k^2).
Showing 1-9 of 9 results.

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