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.

Previous Showing 11-20 of 28 results. Next

A013684 Continued fraction for zeta(8).

Original entry on oeis.org

1, 245, 3, 1, 8, 4, 2, 3, 2, 1, 1, 4, 1, 3, 12, 2, 2, 34, 1, 1, 1, 1, 4, 9, 1, 56, 3, 38, 1, 1, 6, 1, 1, 1, 1, 3, 2, 1, 1, 5, 9, 3, 1, 11, 2, 3, 1, 5, 2, 2, 1, 4, 1, 27, 2, 2, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 72, 17, 1, 36, 1, 5, 6, 1, 4, 10, 1, 4, 1, 4, 1, 1, 1, 8
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A013666 (decimal expansion).
Cf. continued fractions for zeta(2)-zeta(20): A013679, A013631, A013680-A013696.

Programs

  • Mathematica
    ContinuedFraction[Zeta[8],80] (* Harvey P. Dale, Aug 14 2020 *)

Extensions

Offset changed by Andrew Howroyd, Jul 09 2024

A013685 Continued fraction for zeta(9).

Original entry on oeis.org

1, 497, 1, 10, 5, 1, 1, 8, 3, 2, 2, 1, 2, 1, 2, 5, 4, 2, 49, 1, 3, 3, 1, 1, 2, 1, 2, 30, 4, 1, 17, 3, 8, 2, 1, 2, 1, 1, 10, 6, 9, 2, 3, 1, 22, 1, 2, 1, 1, 2, 1, 1, 2, 18, 1, 1, 1, 9, 1, 2, 9, 1, 5, 2, 4, 1, 5, 1, 2, 2, 2, 6, 1, 8, 1, 5, 1, 4, 1483, 1, 3, 1, 2, 7
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A013667 (decimal expansion).
Cf. continued fractions for zeta(2)-zeta(20): A013679, A013631, A013680-A013696.

Programs

  • Mathematica
    ContinuedFraction[Zeta[9],100] (* Harvey P. Dale, Aug 05 2023 *)

Extensions

Offset changed by Andrew Howroyd, Jul 09 2024

A013695 Continued fraction for zeta(19).

Original entry on oeis.org

1, 524050, 1, 1, 2, 3, 1, 1, 1, 1, 3, 1, 2, 5, 14, 1, 5, 1, 3, 1, 3, 1, 3, 2, 1, 1, 1, 4, 1, 9, 1, 2, 1, 6, 2, 1, 1, 1, 76, 85, 1, 8, 1, 1, 7, 12, 7, 2, 1, 2, 4, 1, 3, 1, 22, 1, 3, 6, 1, 1, 1, 1, 1, 4, 1, 11, 1, 3, 1, 2, 1, 6, 1, 9, 1, 2
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A013677 (decimal expansion).
Cf. continued fractions for zeta(2)-zeta(20): A013679, A013631, A013680-A013696.

Programs

  • Mathematica
    ContinuedFraction[Zeta[19], 100] (* Paolo Xausa, Jul 03 2024 *)

Extensions

Offset changed by Andrew Howroyd, Aug 10 2024

A070985 Number of terms in the simple continued fraction for Sum_{k=1..n} 1/k^2.

Original entry on oeis.org

1, 2, 5, 7, 9, 7, 10, 20, 18, 14, 22, 19, 18, 24, 26, 24, 30, 30, 28, 37, 25, 30, 35, 35, 34, 38, 47, 52, 49, 54, 40, 49, 49, 69, 57, 67, 78, 67, 67, 68, 67, 64, 65, 86, 76, 81, 92, 79, 83, 83, 95, 82, 85, 80, 84, 95, 92, 91, 121, 105, 100, 108, 111, 109, 118, 105, 110, 88
Offset: 1

Views

Author

Benoit Cloitre, May 18 2002

Keywords

Comments

Sum_{k>=1} 1/k^2 = zeta(2) = Pi^2/6.

Examples

			The simple continued fraction for Sum_{k=1..10} 1/k^2 is [1, 1, 1, 4, 1, 1, 10, 4, 1, 2, 5, 2, 1, 24] which contains 14 terms, hence a(10) = 14.

Links

Crossrefs

Cf. A013679, A055573, A007406, A007407.

Programs

  • Mathematica
    lcf[f_] := Length[ContinuedFraction[f]]; lcf /@ Accumulate[Table[1/k^2, {k, 1, 100}]] (* Amiram Eldar, Apr 30 2022 *)
  • PARI
    for(n=1,100,print1(length(contfrac(sum(i=1,n,1/i^2))),","))

Formula

Limit_{n ->infinity} a(n)/n = C =1.6....

A013686 Continued fraction for zeta(10).

Original entry on oeis.org

1, 1005, 2, 4, 1, 98, 7, 11, 2, 1, 1, 6, 2, 3, 28, 1, 37, 1, 2, 7, 9, 13, 85, 4, 3, 34, 5, 3, 7, 4, 7, 1, 3, 2, 1, 22, 1, 1, 1, 1, 3, 15, 1, 9, 12, 1, 3, 3, 3, 1, 3, 2, 1, 2, 1, 1, 2, 10, 8, 2, 2, 11, 54, 4, 5, 1, 2, 2, 1, 3, 2, 1, 19, 4, 5, 1, 2, 2, 7, 1, 200
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A013668 (decimal expansion).
Cf. continued fractions for zeta(2)-zeta(20): A013679, A013631, A013680-A013696.

Programs

  • Mathematica
    ContinuedFraction[Zeta[10], 100] (* Paolo Xausa, Jul 03 2024 *)

Extensions

Offset changed by Andrew Howroyd, Jul 09 2024

A013688 Continued fraction for zeta(12).

Original entry on oeis.org

1, 4063, 1, 1, 1, 1, 3, 14, 4, 5, 1, 8, 3, 1, 142, 1, 2, 1, 2, 2, 24, 1, 3, 20, 1, 1, 1, 60, 4, 1, 12, 1, 1, 1, 4, 23, 12, 1, 3, 11, 1, 2, 1, 13, 3, 16, 1, 91, 2, 2, 8, 1, 1, 1, 62, 1, 7, 1, 2, 15, 2, 5, 4, 1, 8, 1, 1, 20, 2, 2, 1, 1, 3, 2, 1, 1, 1, 1, 2, 1, 1
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A013670 (decimal expansion).
Cf. continued fractions for zeta(2)-zeta(20): A013679, A013631, A013680-A013696.

Programs

  • Mathematica
    ContinuedFraction[Zeta[12], 100] (* Paolo Xausa, Jul 03 2024 *)

Extensions

Offset changed by Andrew Howroyd, Jul 09 2024

A013694 Continued fraction for zeta(18).

Original entry on oeis.org

1, 261965, 1, 2, 1, 1, 1, 4, 3, 4, 3, 1, 1, 2, 1, 2, 2, 1, 1, 1, 1, 1, 1, 4, 1, 3, 3, 1, 2, 5, 1, 2, 3, 6, 1, 8, 3, 9, 1, 1, 1, 2, 3, 1, 1, 2, 1, 3, 1, 1, 1, 9, 1, 1, 21, 1, 3, 1, 3, 3, 3, 1, 1, 6, 4, 3, 5, 2, 1, 1, 26, 21, 6, 1, 11, 24, 1
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A013676 (decimal expansion).
Cf. continued fractions for zeta(2)-zeta(20): A013679, A013631, A013680-A013696.

Programs

  • Mathematica
    ContinuedFraction[Zeta[18], 100] (* Paolo Xausa, Jul 03 2024 *)

Extensions

Offset changed by Andrew Howroyd, Aug 10 2024

A322557 Smallest k such that floor(N*sqrt(Sum_{m=1..k} 6/m^2)) = floor(N*Pi), where N = 10^n.

Original entry on oeis.org

7, 23, 600, 1611, 10307, 359863, 1461054, 17819245, 266012440, 1619092245, 10634761313, 97509078554, 1203836807622, 10241799698090, 294871290395291, 4004525174270251, 24827457879988026, 112840588371964574, 2064072875704476882, 15243903003939891921
Offset: 0

Views

Author

Zachary Russ, Aug 28 2019

Keywords

Comments

6*A007406(k)/A007407(k) = Sum_{m=1..k} 6/m^2.
It seems nearly certain that, for all n >= 0, a(n) = ceiling(z - 1/2 - 1/(12*z)) where z = 6/(Pi^2 - (floor(Pi*10^n)/10^n)^2). - Jon E. Schoenfield, Aug 31 2019

Examples

			floor((10^0)*sqrt(Sum_{m=1..7} 6/m^2)) = 3.
floor((10^1)*sqrt(Sum_{m=1..23} 6/m^2)) = 31.
floor((10^2)*sqrt(Sum_{m=1..600} 6/m^2)) = 314.
floor((10^3)*sqrt(Sum_{m=1..1611} 6/m^2)) = 3141.
floor((10^4)*sqrt(Sum_{m=1..10307} 6/m^2)) = 31415.
floor((10^5)*sqrt(Sum_{m=1..359863} 6/m^2)) = 314159.

Links

Crossrefs

Cf. A000796, A013661, A013679, A002388, A274982, A007406, A007407.
Cf. A011545 (floor(Pi*10^n)).

Programs

  • PARI
    a(n) = {my(k = 1); t = floor(10^(n)*Pi); while(floor(10^(n)*sqrt(sum(m = 1, k, 6/m^2))) != t, k++); k; } \\ Jinyuan Wang, Aug 30 2019

Extensions

a(6)-a(19) from Jon E. Schoenfield, Aug 31 2019

A013687 Continued fraction for zeta(11).

Original entry on oeis.org

1, 2023, 1, 1, 12, 1, 2, 2, 1, 102, 1, 44, 1, 2, 2, 1, 2, 3, 1, 5, 2, 1, 1, 2, 1, 13, 4, 14, 2, 5, 1, 5, 1, 6, 1, 2, 9, 1, 1, 1, 1, 7, 1, 2, 3, 1, 39, 3, 119, 12, 1, 1, 5, 1, 1, 151, 3, 4, 1, 2, 4, 98, 29, 6, 2, 1, 3, 9, 1, 1, 1, 5, 1, 2
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A013669.
Cf. continued fractions for zeta(2)-zeta(20): A013679, A013631, A013680-A013696.

Programs

  • Mathematica
    ContinuedFraction[Zeta[11],80] (* Harvey P. Dale, May 22 2013 *)

Extensions

Offset changed by Andrew Howroyd, Jul 08 2024

A013689 Continued fraction for zeta(13).

Original entry on oeis.org

1, 8149, 13, 1, 2, 1, 6, 23, 3, 1, 7, 1, 1, 5, 1, 1, 4, 1, 1, 1, 4, 1, 1, 2, 2, 8, 1, 29, 32, 22, 2, 123, 1, 2, 1, 10, 1, 2, 2, 1, 4, 1, 13, 5, 8, 34, 2, 1, 7, 1, 2, 1, 3, 20, 8, 1, 4, 1, 5, 1, 59, 3, 7, 1, 1, 3, 2, 6, 1, 1, 2, 9, 1, 1
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A013671.
Cf. continued fractions for zeta(2)-zeta(20): A013679, A013631, A013680-A013696.

Programs

  • Mathematica
    ContinuedFraction[Zeta[13],100] (* Harvey P. Dale, Feb 25 2015 *)

Extensions

Offset changed by Andrew Howroyd, Jul 08 2024
Previous Showing 11-20 of 28 results. Next

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

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