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 1-5 of 5 results.

A081573 Decimal expansion of Sum_(1/(2^q-1)) with the summation extending over all pairs of integers gcd(p,q) = 1, 0 < p/q < e = 2.718... .

Original entry on oeis.org

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

Views

Author

Benoit Cloitre, Apr 21 2003

Keywords

Examples

			4.866141732...

Links

Crossrefs

Cf. A001113 (e).
Cf. A081544, A081550, A081564.

Programs

  • Mathematica
    With[{digmax = 120}, RealDigits[Sum[1/2^Floor[k/E], {k, 1, 20*digmax}], 10, digmax][[1]]] (* Amiram Eldar, May 25 2023 *)

Formula

Equals Sum_{k>=1} (1/2)^floor(k/e) = Sum_{k>=1} 1/2^A032634(k).

Extensions

Data corrected by Amiram Eldar, May 25 2023

A081544 Decimal expansion of Sum_(1/(2^q-1)) with the summation extending over all pairs of integers gcd(p,q) = 1, 0 < p/q < phi, where phi is the Golden ratio.

Original entry on oeis.org

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

Views

Author

Benoit Cloitre, Apr 21 2003

Keywords

Links

Crossrefs

Cf. A001622 (golden ratio), A014565, A073115.
Cf. A081550, A081564, A081573.

Programs

  • Mathematica
    With[{digmax = 120}, RealDigits[Sum[1/2^Floor[k/GoldenRatio], {k, 1, 10*digmax}], 10, digmax][[1]]] (* Amiram Eldar, May 25 2023 *)

Formula

Equals Sum_{k>=1} (1/2)^floor(k/phi).
Equals A014565 + 2 = A073115 + 1. - Amiram Eldar, May 25 2023

Extensions

Data corrected by Amiram Eldar, May 25 2023

A081550 Decimal expansion of Sum_(1/(2^q-1)) with the summation extending over all pairs of integers gcd(p,q) = 1, 0 < p/q < Pi.

Original entry on oeis.org

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

Views

Author

Benoit Cloitre, Apr 21 2003

Keywords

Examples

			6.007874015...

Links

Crossrefs

Cf. A000796 (Pi).
Cf. A081544, A081564, A081573.

Programs

  • Mathematica
    With[{digmax = 120}, RealDigits[Sum[1/2^Floor[k/Pi], {k, 1, 20*digmax}], 10, digmax][[1]]] (* Amiram Eldar, May 25 2023 *)

Formula

Equals Sum_{k>=1} (1/2)^floor(k/Pi) = Sum_{k>=1} 1/2^A032615(k).

Extensions

Data corrected by Amiram Eldar, May 25 2023

A081565 Binomial transform of expansion of exp(3cosh(x)).

Original entry on oeis.org

1, 1, 4, 10, 49, 181, 1039, 4915, 32134, 182206, 1330609, 8706655, 70012309, 515822581, 4517489344, 36835737130, 348313165249, 3103526872081, 31462900577419, 303344232041215, 3277823503679554, 33930282904263406
Offset: 0

Views

Author

Paul Barry, Mar 22 2003

Keywords

Comments

Binomial transform of A081564.

Links

Crossrefs

Cf. A081564, A081565.

Programs

  • Magma
    m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp(3*Cosh(x)+x-3) )); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Aug 13 2019
  • Maple
    seq(coeff(series(exp(3*cosh(x)+x-3), x, n+1)*factorial(n), x, n), n = 0 .. 30); # G. C. Greubel, Aug 13 2019
  • Mathematica
    With[{nn = 30}, CoefficientList[Series[Exp[3 Cosh[x] + x - 3], {x, 0, nn}], x] Range[0, nn]!] (* Vincenzo Librandi, Aug 08 2013 *)
  • PARI
    my(x='x+O('x^30)); Vec(serlaplace( exp(3*cosh(x)+x-3) )) \\ G. C. Greubel, Aug 13 2019
  • Sage
    [factorial(n)*( exp(3*cosh(x)+x-3) ).series(x,n+1).list()[n] for n in (0..30)] # G. C. Greubel, Aug 13 2019

Formula

E.g.f.: exp(x) * exp(3*cosh(x))/e^3 = exp(3*cosh(x)+x-3).

A081566 Second binomial transform of expansion of exp(3cosh(x)).

Original entry on oeis.org

1, 2, 7, 26, 118, 572, 3127, 18146, 114793, 765602, 5463982, 40870436, 323326813, 2667777842, 23092966267, 207651618746, 1947316349278, 18906249136892, 190564801592107, 1982986181092226, 21345005629846213, 236628248493001202
Offset: 0

Views

Author

Paul Barry, Mar 22 2003

Keywords

Comments

Binomial transform of A081565.

Links

Crossrefs

Cf. A081564, A081565.

Programs

  • Magma
    m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp(3*Cosh(x)+2*x-3) )); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Aug 13 2019
  • Maple
    seq(coeff(series(exp(3*cosh(x)+2*x-3), x, n+1)*factorial(n), x, n), n = 0 .. 30); # G. C. Greubel, Aug 13 2019
  • Mathematica
    With[{nn = 30}, CoefficientList[Series[Exp[3 Cosh[x] + 2 x - 3], {x, 0, nn}], x] Range[0, nn]!] (* Vincenzo Librandi, Aug 08 2013 *)
  • PARI
    my(x='x+O('x^30)); Vec(serlaplace( exp(3*cosh(x)+2*x-3) )) \\ G. C. Greubel, Aug 13 2019
  • Sage
    [factorial(n)*( exp(3*cosh(x)+2*x-3) ).series(x,n+1).list()[n] for n in (0..30)] # G. C. Greubel, Aug 13 2019

Formula

E.g.f.: exp(2*x) * exp(3*cosh(x))/e^3 = exp(3*cosh(x)+2*x-3).
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.