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 1-10 of 11 results. Next

A077635 LCM of terms in periodic part of continued fraction expansion of square root of -1+3^n.

Original entry on oeis.org

2, 4, 10, 16, 210, 52, 10764, 160, 840, 484, 78225840, 1456, 5729631692400, 4372, 932723933480442148565520, 13120, 202096081896183783466278120, 39364, 1505075187143521116689096930825555405884185492185368021027531191578186233461600
Offset: 1

Views

Author

Labos Elemer, Nov 13 2002

Keywords

Links

Crossrefs

Cf. A077624-A077635.

Programs

  • Mathematica
    Table[Apply[LCM, Last[ContinuedFraction[Sqrt[ -1+3^u]]]], {u, 1, 25}]

A077625 Largest term in periodic part of continued fraction expansion of square root of -1+2^n or 0 if -1+2^n is square.

Original entry on oeis.org

0, 2, 4, 6, 10, 14, 22, 30, 44, 62, 90, 126, 180, 254, 362, 510, 724, 1022, 1448, 2046, 2896, 4094, 5792, 8190, 11584, 16382, 23170, 32766, 46340, 65534, 92680, 131070, 185362, 262142, 370726, 524286, 741454, 1048574, 1482910, 2097150, 2965820, 4194302, 5931640
Offset: 1

Views

Author

Labos Elemer, Nov 13 2002

Keywords

Comments

a(n) = 0 iff n = 1, as a consequence of Catalan's conjecture or Mihăilescu's theorem. - Bernard Schott, Apr 24 2022

Links

Crossrefs

Cf. A077624-A077635.

Programs

  • Mathematica
    Table[Max[Last[ContinuedFraction[Sqrt[ -1+2^u]]]], {u, 1, 32}]

Extensions

Definition clarified, a(1) corrected and a(33)-a(43) added by Chai Wah Wu, Apr 19 2022

A077631 Sum of terms in periodic part of continued fraction expansion of square root of -1 + 3^n.

Original entry on oeis.org

2, 5, 10, 17, 56, 53, 160, 161, 346, 485, 1850, 1457, 3764, 4373, 13468, 13121, 43572, 39365, 192642, 118097, 226348, 354293, 2646006, 1062881, 1871694, 3188645, 5646564, 9565937, 36393508, 28697813, 143274092, 86093441, 195407590, 258280325, 542628818, 774840977
Offset: 1

Views

Author

Labos Elemer, Nov 13 2002

Keywords

Links

Crossrefs

Cf. A077624-A077635.

Programs

  • Mathematica
    Table[Apply[Plus, Last[ContinuedFraction[Sqrt[ -1+3^u]]]], {u, 1, 25}]

Formula

From Benoit Cloitre, Nov 28 2002: (Start)
a(2n) = 2*3^n-1.
Conjecture: if n>1 sqrt(3^(2n+1)) < a(2n+1) < sqrt((2n+1)*3^(2n+1)). (End)

Extensions

a(26)-a(36) from Chai Wah Wu, Sep 18 2021

A077626 Largest term in periodic part of continued fraction expansion of square root of 1+3^n or 0 if 1+3^n is square.

Original entry on oeis.org

0, 6, 10, 18, 30, 54, 92, 162, 280, 486, 840, 1458, 2524, 4374, 7574, 13122, 22726, 39366, 68182, 118098, 204550, 354294, 613654, 1062882, 1840964, 3188646, 5522896, 9565938, 16568690, 28697814, 49706070, 86093442, 149118214, 258280326, 447354646, 774840978
Offset: 1

Views

Author

Labos Elemer, Nov 13 2002

Keywords

Comments

a(n) = 0 iff n = 1, as a consequence of Catalan's conjecture or Mihăilescu's theorem. - Bernard Schott, Apr 25 2022

Crossrefs

Cf. A077624-A077635.
Equals 2*A017913(n) for n > 1.

Programs

  • Mathematica
    Table[Max[Last[ContinuedFraction[Sqrt[1+3^u]]]], {u, 1, 32}]
  • PARI
    a(n) = if (n==1, 0, 2*sqrtint(3^n)); \\ Michel Marcus, Apr 20 2022

Extensions

a(1) changed and definition clarified by Chai Wah Wu, Sep 18 2021
a(31)-a(36) from Chai Wah Wu, Apr 20 2022

A077627 Largest term in periodic part of continued fraction expansion of square root of -1+3^n.

Original entry on oeis.org

2, 4, 10, 16, 30, 52, 92, 160, 280, 484, 840, 1456, 2524, 4372, 7574, 13120, 22726, 39364, 68182, 118096, 204550, 354292, 613654, 1062880, 1840964, 3188644, 5522896, 9565936, 16568690, 28697812, 49706070, 86093440, 149118214, 258280324, 447354646, 774840976, 1342063940
Offset: 1

Views

Author

Labos Elemer, Nov 13 2002

Keywords

Links

Crossrefs

Cf. A077631, A077635.
Cf. A077624, A077625, A077626.

Programs

  • Mathematica
    Table[Max[Last[ContinuedFraction[Sqrt[ -1+3^u]]]], {u, 1, 32}]

Formula

a(2*m) = 2*(3^m-1); in general a(n) is close to 2*(3^(n/2)-1) and for any n, 0 <= a(n) - 2*(3^(n/2)-1) < 2. Conjecture: a(n)=ceiling(2*(3^(n/2)-1)) except for n=3, 9, 27 and all powers of 3, in this case a(n)=1+ceiling(2*(3^(n/2)-1)). - Benoit Cloitre, Nov 24 2002

Extensions

a(31)-a(37) from Chai Wah Wu, Oct 01 2019

A077628 Sum of terms in periodic part of continued fraction expansion of square root of 1+2^n.

Original entry on oeis.org

3, 4, 0, 8, 14, 16, 42, 32, 82, 64, 270, 128, 518, 256, 642, 512, 2126, 1024, 7098, 2048, 7502, 4096, 14550, 8192, 19810, 16384, 108802, 32768, 101154, 65536, 299166, 131072, 360298, 262144, 891666, 524288, 2128896, 1048576, 8352642, 2097152, 4466878
Offset: 1

Views

Author

Labos Elemer, Nov 13 2002

Keywords

Links

Crossrefs

Cf. A077624-A077635.

Programs

  • Mathematica
    Table[Apply[Plus, Last[ContinuedFraction[Sqrt[1+2^u]]]], {u, 1, 25}]
Join[{3,4,3},Table[Total[ContinuedFraction[Sqrt[1+2^n]][[2]]],{n,4,50}]] (* Harvey P. Dale, Oct 03 2016 *)

Extensions

More terms from Harvey P. Dale, Oct 03 2016
a(3) corrected by Chai Wah Wu, Sep 18 2021

A077629 Sum of terms in periodic part of continued fraction expansion of square root of -1+2^n.

Original entry on oeis.org

0, 3, 7, 7, 25, 15, 63, 31, 125, 63, 187, 127, 1081, 255, 1091, 511, 1247, 1023, 11217, 2047, 11999, 4095, 20259, 8191, 66077, 16383, 88609, 32767, 55491, 65535, 284231, 131071, 413199, 262143, 585102, 524287, 3351311, 1048575, 3462843, 2097151
Offset: 1

Views

Author

Labos Elemer, Nov 13 2002

Keywords

Links

Crossrefs

Cf. A077624-A077635.

Programs

  • Mathematica
    Table[Apply[Plus, Last[ContinuedFraction[Sqrt[ -1+2^u]]]], {u, 1, 25}]

Formula

a(2*n) = 2^(n+1) - 1. - Max Alekseyev, May 13 2010

Extensions

a(1) corrected, a(26) and on added by Max Alekseyev, May 13 2010

A077630 Sum of terms in periodic part of continued fraction expansion of square root of 1+3^n.

Original entry on oeis.org

0, 6, 18, 18, 86, 54, 344, 162, 650, 486, 3246, 1458, 11542, 4374, 19196, 13122, 83672, 39366, 434242, 118098, 248278, 354294, 5669112, 1062882, 10522426, 3188646, 5969216, 9565938, 46110338, 28697814, 271512140, 86093442, 149401156, 258280326, 583061114, 774840978
Offset: 1

Views

Author

Labos Elemer, Nov 13 2002

Keywords

Comments

a(1) = 0 since 3^1+1=4 is a square and sqrt(4) = 2 has no periodic part in its continued fraction expansion. - Chai Wah Wu, Sep 17 2021

Links

Crossrefs

Cf. A077624-A077635.

Programs

  • Mathematica
    Table[Apply[Plus, Last[ContinuedFraction[Sqrt[1+3^u]]]], {u, 1, 25}]

Extensions

a(1) corrected and a(26)-a(36) added by Chai Wah Wu, Sep 17 2021

A077632 LCM of terms in periodic part of continued fraction expansion of square root of 1+2^n.

Original entry on oeis.org

2, 4, 3, 8, 10, 16, 66, 32, 220, 64, 235620, 128, 360360, 256, 2508660, 512, 16290726273360, 1024, 23508390120198798054240, 2048, 465977246353354335600, 4096, 3026207803243202829468977760, 8192
Offset: 1

Views

Author

Labos Elemer, Nov 13 2002

Keywords

Crossrefs

Cf. A077624-A077635.

Programs

  • Mathematica
    Table[Apply[LCM, Last[ContinuedFraction[Sqrt[1+2^u]]]], {u, 1, 25}]

A077633 LCM of terms in periodic part of continued fraction expansion of square root of -1+2^n.

Original entry on oeis.org

1, 2, 4, 6, 30, 14, 462, 30, 924, 62, 360, 126, 88623545610600, 254, 5364840, 510, 214057116, 1022, 473045752778491241949600, 2046, 3968969477588889464597306400, 4094, 16579321550270765937400453704000
Offset: 1

Views

Author

Labos Elemer, Nov 13 2002

Keywords

Crossrefs

Cf. A077624-A077635.

Programs

  • Mathematica
    Table[Apply[LCM, Last[ContinuedFraction[Sqrt[ -1+2^u]]]], {u, 1, 25}]
Showing 1-10 of 11 results. Next

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