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 1-6 of 6 results.

A248308 Egyptian fraction representation of sqrt(85) (A010536) using a greedy function.

Original entry on oeis.org

9, 5, 52, 3188, 84918152, 16076240863422804, 414301661246661635838512408392212, 322874955397009416384181733402291919609578193183044079681678443089, 162879628085209618118393406431670411111270528217363349375030319925289267303145152426058897418548846768803944306076778632776827619214
Offset: 0

Views

Author

Robert G. Wilson v, Oct 04 2014

Keywords

Crossrefs

Egyptian fraction representations of the square roots: A006487, A224231, A248235-A248322.
Egyptian fraction representations of the cube roots: A129702, A132480-A132574.

Programs

  • Mathematica
    Egyptian[nbr_] := Block[{lst = {IntegerPart[nbr]}, cons = N[ FractionalPart[ nbr], 2^20], denom, iter = 8}, While[ iter > 0, denom = Ceiling[ 1/cons]; AppendTo[ lst, denom]; cons -= 1/denom; iter--]; lst]; Egyptian[ Sqrt[ 85]]

A041150 Numerators of continued fraction convergents to sqrt(85).

Original entry on oeis.org

9, 37, 46, 83, 378, 6887, 27926, 34813, 62739, 285769, 5206581, 21112093, 26318674, 47430767, 216041742, 3936182123, 15960770234, 19896952357, 35857722591, 163327842721, 2975758891569, 12066363408997
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

From Johannes W. Meijer, Jun 17 2010: (Start)
The a(n) terms of this sequence can be constructed with the terms of sequence A087798.
For the terms of the periodic sequence of the continued fraction for sqrt(85) see A010158. We observe that its period is five. The decimal expansion of sqrt(85) is A010536. (End)

Links

Crossrefs

Cf. A010536, A041018, A041046, A041090, A041150, A041151, A041226, A041318, A041426,
A041550.

Programs

  • Mathematica
    Numerator[Convergents[Sqrt[85], 30]] (* Vincenzo Librandi, Oct 29 2013 *)

Formula

From Johannes W. Meijer, Jun 17 2010: (Start)
a(5*n) = A087798(3*n+1), a(5*n+1) = (A087798(3*n+2) - A087798(3*n+1))/2, a(5*n+2) = (A087798(3*n+2) + A087798(3*n+1))/2, a(5*n+3) = A087798(3*n+2) and a(5*n+4) = A087798(3*n+3)/2. (End)
G.f.: -(x^9-9*x^8+37*x^7-46*x^6+83*x^5+378*x^4+83*x^3+46*x^2+37*x+9) / (x^10+756*x^5-1). - Colin Barker, Nov 04 2013

A041151 Denominators of continued fraction convergents to sqrt(85).

Original entry on oeis.org

1, 4, 5, 9, 41, 747, 3029, 3776, 6805, 30996, 564733, 2289928, 2854661, 5144589, 23433017, 426938895, 1731188597, 2158127492, 3889316089, 17715391848, 322766369353, 1308780869260, 1631547238613, 2940328107873, 13392859670105, 244011802169763, 989440068349157
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

From Johannes W. Meijer, Jun 12 2010: (Start)
The a(n) terms of this sequence can be constructed with the terms of sequence A099371.
For the terms of the periodic sequence of the continued fraction for sqrt(85) see A010158. We observe that its period is five. The decimal expansion of sqrt(85) is A010536. (End)

Links

Crossrefs

Cf. A041150, A041019, A041047, A041091, A041151, A041227, A041319, A041427, A041551.

Programs

  • Magma
    I:=[1, 4, 5, 9, 41, 747, 3029, 3776, 6805, 30996]; [n le 10 select I[n] else 756*Self(n-5)+Self(n-10): n in [1..30]]; // Vincenzo Librandi, Dec 12 2013
  • Mathematica
    Table[Denominator[FromContinuedFraction[ContinuedFraction[Sqrt[85], n]]], {n, 1, 50}] (* Vladimir Joseph Stephan Orlovsky, Jun 23 2011 *)
Denominator[Convergents[Sqrt[85], 30]] (* Vincenzo Librandi, Dec 12 2013 *)

Formula

From Johannes W. Meijer, Jun 12 2010: (Start)
a(5*n) = A099371(3*n+1), a(5*n+1) = (A099371(3*n+2)-A099371(3*n+1))/2, a(5*n+2) = (A099371(3*n+2)+A099371(3*n+1))/2, a(5*n+3):= A099371(3*n+2) and a(5*n+4) = A099371(3*n+3)/2. (End)
G.f.: -(x^8-4*x^7+5*x^6-9*x^5+41*x^4+9*x^3+5*x^2+4*x+1) / (x^10+756*x^5-1). - Colin Barker, Nov 11 2013
a(n) = 756*a(n-5) + a(n-10). - Vincenzo Librandi, Dec 12 2013

A176522 Decimal expansion of (9+sqrt(85))/2.

Original entry on oeis.org

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

Views

Author

Klaus Brockhaus, Apr 23 2010

Keywords

Comments

Continued fraction expansion of (9+sqrt(85))/2 is A010734.

Examples

			(9+sqrt(85))/2 = 9.10977222864644365500...

Links

Crossrefs

Cf. A010536 (decimal expansion of sqrt(85)), A010734 (all 9's sequence), A333345, A049310.

Programs

Formula

Equals lim_{n->infinity} S(n, sqrt(5*17))/S(n-1, sqrt(5*17)), with the S-Chebyshev polynomials (see A049310). - Wolfdieter Lang, Nov 15 2023

A010158 Continued fraction for sqrt(85).

Original entry on oeis.org

9, 4, 1, 1, 4, 18, 4, 1, 1, 4, 18, 4, 1, 1, 4, 18, 4, 1, 1, 4, 18, 4, 1, 1, 4, 18, 4, 1, 1, 4, 18, 4, 1, 1, 4, 18, 4, 1, 1, 4, 18, 4, 1, 1, 4, 18, 4, 1, 1, 4, 18, 4, 1, 1, 4, 18, 4, 1, 1, 4, 18, 4, 1, 1, 4, 18, 4, 1, 1, 4, 18, 4, 1, 1, 4
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Examples

			9.219544457292887310002274281... = 9 + 1/(4 + 1/(1 + 1/(1 + 1/(4 + ...)))). - _Harry J. Smith_, Jun 10 2009

Links

Crossrefs

Cf. A010536 Decimal expansion. - Harry J. Smith, Jun 10 2009

Programs

  • Mathematica
    ContinuedFraction[Sqrt[85],300] (* Vladimir Joseph Stephan Orlovsky, Mar 09 2011 *)
  • PARI
    { allocatemem(932245000); default(realprecision, 24000); x=contfrac(sqrt(85)); for (n=0, 20000, write("b010158.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 10 2009

A177347 Decimal expansion of (5+sqrt(85))/10.

Original entry on oeis.org

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

Views

Author

Klaus Brockhaus, May 07 2010

Keywords

Comments

Continued fraction expansion of (5+sqrt(85))/10 is A130196.

Examples

			(5+sqrt(85))/10 = 1.42195444572928873100...

Crossrefs

Cf. A010536 (decimal expansion of sqrt(85)), A130196 (repeat 1, 2, 2).

Programs

  • Mathematica
    RealDigits[(5+Sqrt[85])/10,10,120][[1]] (* Harvey P. Dale, Dec 31 2011 *)
Showing 1-6 of 6 results.

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

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