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 1-4 of 4 results.

A059866 Period of the continued fraction for sqrt(2^n-1).

Original entry on oeis.org

2, 4, 2, 8, 2, 12, 2, 20, 2, 12, 2, 164, 2, 40, 2, 40, 2, 1208, 2, 660, 2, 1304, 2, 3056, 2, 2492, 2, 1080, 2, 13004, 2, 10232, 2, 11296, 2, 148736, 2, 56576, 2, 615482, 2, 44448, 2, 64, 2, 2628524, 2, 28219952, 2, 139558, 2, 3067080, 2, 2683626, 2, 90740360, 2, 103050292, 2
Offset: 2

Views

Author

Labos Elemer, Feb 28 2001

Keywords

Examples

			For n=7 and n=8 the continued fractions are [[11], [3, 1, 2, 2, 7, 11, 7, 2, 2, 1, 3, 22]] and [[15], [1, 30]] with periods 12 and 2, respectively.

Links

Crossrefs

Cf. A003285, A059926, A059927, A064932.

Programs

  • Maple
    with(numtheory): [seq(nops(cfrac(sqrt(2^k-1),'periodic','quotients')[2]),k=2..30)];
  • Mathematica
    Table[Length@ Last@ ContinuedFraction[Sqrt[2^n - 1]], {n, 2, 56}] (* Michael De Vlieger, Mar 21 2015 *)

Extensions

Corrected and extended by Naohiro Nomoto, Nov 09 2001
a(57)-a(60) from Daniel Suteu, Jan 25 2019

A061682 Length of period of continued fraction expansion of square root of (2^(2n+1)+1).

Original entry on oeis.org

4, 10, 16, 44, 74, 46, 204, 714, 702, 908, 404, 7754, 1136, 9886, 8154, 23578, 65096, 404762, 23992, 3514774, 110124, 4802160, 6490450, 180832, 115972, 770304, 62665998, 133093360, 1019300318, 60079334, 113987888, 5702124038, 4463754028, 713372392, 38574516706, 9096543466, 7030527700, 582442851838, 16708770664, 32628786870
Offset: 2

Views

Author

Labos Elemer, Mar 01 2001

Keywords

Comments

Old definition was: "Quotient cycle length in continued fraction expansion of sqrt(2^(2n+1)+1)."

Crossrefs

Cf. A003285, A059866, A059926, A087289.

Programs

  • Mathematica
    a[n_] := ContinuedFraction[Sqrt[2^(2n+1)+1]] // Last // Length; Table[a[n], {n, 2, 28}] (* Jean-François Alcover, Dec 11 2016 *)

Formula

a(n) = A003285(A087289(n)). - Michel Marcus, Sep 26 2019

Extensions

One more term from David W. Wilson, Jun 18 2001
Corrected and extended by Naohiro Nomoto, Nov 09 2001
a(29)-a(31) from Daniel Suteu, Jan 25 2019
a(32) from Chai Wah Wu, Sep 23 2019
a(33)-a(38) from Chai Wah Wu, Sep 25 2019
Simpler definition from Bernard Schott, Sep 26 2019
a(39)-a(41) from Chai Wah Wu, Sep 29 2019

A062328 Length of period of continued fraction expansion of square root of 3^n+1.

Original entry on oeis.org

1, 0, 1, 4, 1, 26, 1, 56, 1, 44, 1, 264, 1, 814, 1, 136, 1, 3730, 1, 20968, 1, 2448, 1, 287980, 1, 397238, 1, 2678, 1, 670896, 1, 8110044, 1, 20696, 1, 1066520, 1, 366601254, 1, 277444, 1, 5903828476, 1, 7701738148, 1, 8208058, 1, 30287795640, 1, 253244432640, 1, 11656644672, 1, 2376211301858, 1, 590009437260, 1
Offset: 0

Views

Author

Labos Elemer, Jul 13 2001

Keywords

Comments

a(n) = 1 iff n is even. In this case, 3^n + 1 = A002522(3^(n/2)) and the continued fraction expansion of sqrt(3^n+1) is {3^(n/2); 2*3^(n/2), 2*3^(n/2), 2*3^(n/2), 2*3^(n/2), ...}. - Bernard Schott, Sep 25 2019

Examples

			The period of sqrt(244) contains 26 terms: [1, 1, 1, 1, 1, 2, 1, 5, 1, 1, 9, 1, 6, 1, 9, 1, 1, 5, 1, 2, 1, 1, 1, 1, 1, 30], so a(5) = 26.

Crossrefs

Cf. A003285, A034472, A059866, A059926, A059927, A062345, A062682.

Programs

  • Maple
    with(numtheory): [seq(nops(cfrac(sqrt(3^k+1),'periodic','quotients')[2]),k=2..18)];
  • Mathematica
    Table[Length[Last[ContinuedFraction[Sqrt[3^w+1]]]],{w,1,40}] (* corrected by Harvey P. Dale, Dec 05 2014 *)

Formula

a(n) = A003285(A034472(n)). - Bernard Schott, Sep 25 2019

Extensions

More terms from Harvey P. Dale, Dec 05 2014
a(41)-a(42) from Vaclav Kotesovec, Sep 17 2019
a(0), a(43)-a(48) from Chai Wah Wu, Sep 25 2019
a(49)-a(56) from Chai Wah Wu, Oct 03 2019

A062345 Length of period of continued fraction expansion of square root of 3^n-1.

Original entry on oeis.org

1, 2, 1, 2, 10, 2, 19, 2, 25, 2, 156, 2, 149, 2, 580, 2, 716, 2, 6461, 2, 2485, 2, 123256, 2, 64, 2, 8638, 2, 722190, 2, 3804214, 2, 1783536, 2, 3550696, 2, 86022946, 2, 22119349, 2, 692630166, 2, 8247763078, 2, 43380360, 2, 15150768502, 2, 10229872316, 2, 36580802370, 2, 333495606762, 2, 676122216162, 2
Offset: 1

Views

Author

Labos Elemer, Jul 13 2001

Keywords

Examples

			The period of sqrt(242) contains 10 terms: [1,1,3,1,14,1,3,1,1,30]

Crossrefs

Cf. A003285, A024023, A059866, A059926, A059927, A062328.

Programs

  • Maple
    with(numtheory): [seq(nops(cfrac(sqrt(3^k-1),'periodic','quotients')[2]),k=1..16)];
  • Mathematica
    Table[Length[Last[ContinuedFraction[Sqrt[ -1+3^u]]]], {u, 1, 36}]

Formula

a(n) = A003285(A024023(n)). - Michel Marcus, Sep 25 2019

Extensions

a(37)-a(42) from Vaclav Kotesovec, Aug 28 2019
a(43)-a(44) from Vaclav Kotesovec, Sep 17 2019
a(45)-a(52) from Chai Wah Wu, Sep 25 2019
a(53)-a(56) from Chai Wah Wu, Sep 29 2019
Showing 1-4 of 4 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.