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.

A248302 Egyptian fraction representation of sqrt(78) (A010530) using a greedy function.

Original entry on oeis.org

8, 2, 4, 13, 207, 145528, 2014567277837, 18506674542689699353989922, 29204169131207852528143087130566230597483060288517588, 1413186736193694972997145387255081607494646716902772853300760690304360053815944509194409970991405502998192
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[ 78]]

A010156 Continued fraction for sqrt(78).

Original entry on oeis.org

8, 1, 4, 1, 16, 1, 4, 1, 16, 1, 4, 1, 16, 1, 4, 1, 16, 1, 4, 1, 16, 1, 4, 1, 16, 1, 4, 1, 16, 1, 4, 1, 16, 1, 4, 1, 16, 1, 4, 1, 16, 1, 4, 1, 16, 1, 4, 1, 16, 1, 4, 1, 16, 1, 4, 1, 16, 1, 4, 1, 16, 1, 4, 1, 16, 1, 4, 1, 16, 1, 4, 1, 16
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Examples

			8.83176086632784685476404272... = 8 + 1/(1 + 1/(4 + 1/(1 + 1/(16 + ...)))). - _Harry J. Smith_, Jun 09 2009

Links

Crossrefs

Cf. A010530 (decimal expansion).

Programs

  • Mathematica
    ContinuedFraction[Sqrt[78],300] (* Vladimir Joseph Stephan Orlovsky, Mar 09 2011 *)
PadRight[{8},120,{16,1,4,1}] (* Harvey P. Dale, Sep 10 2023 *)
  • PARI
    { allocatemem(932245000); default(realprecision, 21000); x=contfrac(sqrt(78)); for (n=0, 20000, write("b010156.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 09 2009

Formula

From Amiram Eldar, Nov 13 2023: (Start)
Multiplicative with a(2) = 4, a(2^e) = 16 for e >= 2, and a(p^e) = 1 for an odd prime p.
Dirichlet g.f.: zeta(s) * (1 + 3/2^s + 4/4^(s-1)). (End)

A041138 Numerators of continued fraction convergents to sqrt(78).

Original entry on oeis.org

8, 9, 44, 53, 892, 945, 4672, 5617, 94544, 100161, 495188, 595349, 10020772, 10616121, 52485256, 63101377, 1062107288, 1125208665, 5562941948, 6688150613, 112573351756, 119261502369, 589619361232, 708880863601, 11931713178848, 12640594042449, 62494089348644
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A010530, A041139.

Programs

  • Mathematica
    Numerator[Convergents[Sqrt[78], 30]] (* Vincenzo Librandi, Oct 29 2013 *)

Formula

G.f.: -(x^7-8*x^6+9*x^5-44*x^4-53*x^3-44*x^2-9*x-8) / (x^8-106*x^4+1). - Colin Barker, Nov 05 2013

Extensions

More terms from Colin Barker, Nov 05 2013

A041139 Denominators of continued fraction convergents to sqrt(78).

Original entry on oeis.org

1, 1, 5, 6, 101, 107, 529, 636, 10705, 11341, 56069, 67410, 1134629, 1202039, 5942785, 7144824, 120259969, 127404793, 629879141, 757283934, 12746422085, 13503706019, 66761246161, 80264952180, 1351000481041, 1431265433221, 7076062213925, 8507327647146
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A041138, A010156, A020835, A010530.

Programs

  • Magma
    I:=[1,1,5,6,101,107,529,636]; [n le 8 select I[n] else 106*Self(n-4)-Self(n-8): n in [1..40]]; // Vincenzo Librandi, Dec 11 2013
  • Mathematica
    Denominator/@Convergents[Sqrt[78], 50] (* Vladimir Joseph Stephan Orlovsky, Jul 05 2011 *)
CoefficientList[Series[-(x^2 - x - 1) (x^4 + 6*x^2 + 1)/(x^8 - 106 x^4 + 1), {x, 0, 30}], x] (* Vincenzo Librandi, Dec 11 2013 *)
LinearRecurrence[{0,0,0,106,0,0,0,-1},{1,1,5,6,101,107,529,636},30] (* Harvey P. Dale, Sep 15 2018 *)

Formula

G.f.: -(x^2-x-1)*(x^4+6*x^2+1) / (x^8-106*x^4+1). - Colin Barker, Nov 13 2013
a(n) = 106*a(n-4) - a(n-8). - Vincenzo Librandi, Dec 11 2013
Showing 1-4 of 4 results.

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

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