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.

A248261 Egyptian fraction representation of sqrt(34) (A010489) using a greedy function.

Original entry on oeis.org

5, 2, 4, 13, 249, 78409, 36737419013, 3360517821921008389676, 12410117686109445240372967020019944131780632, 3346975977981026206584708326983128003661219924365061759193139960235987881485856695085453
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[ 34]]

A010132 Continued fraction for sqrt(34).

Original entry on oeis.org

5, 1, 4, 1, 10, 1, 4, 1, 10, 1, 4, 1, 10, 1, 4, 1, 10, 1, 4, 1, 10, 1, 4, 1, 10, 1, 4, 1, 10, 1, 4, 1, 10, 1, 4, 1, 10, 1, 4, 1, 10, 1, 4, 1, 10, 1, 4, 1, 10, 1, 4, 1, 10, 1, 4, 1, 10, 1, 4, 1, 10, 1, 4, 1, 10, 1, 4, 1, 10, 1, 4, 1, 10
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Examples

			5.83095189484530047087415287... = 5 + 1/(1 + 1/(4 + 1/(1 + 1/(10 + ...)))). - _Harry J. Smith_, Jun 04 2009

References

  • James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 276.

Links

Crossrefs

Cf. A010489 (decimal expansion).

Programs

  • Mathematica
    ContinuedFraction[Sqrt[34],300] (* Vladimir Joseph Stephan Orlovsky, Mar 06 2011 *)
  • PARI
    { allocatemem(932245000); default(realprecision, 19000); x=contfrac(sqrt(34)); for (n=0, 20000, write("b010132.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 04 2009

Formula

From Amiram Eldar, Nov 12 2023: (Start)
Multiplicative with a(2) = 4, a(2^e) = 10 for e >= 2, and a(p^e) = 1 for an odd prime p.
Dirichlet g.f.: zeta(s) * (1 + 3/2^(2*s-1) + 3/2^s). (End)
G.f.: (5 + x + 4*x^2 + x^3 + 5*x^4)/(1 - x^4). - Stefano Spezia, Jul 27 2025

A041056 Numerators of continued fraction convergents to sqrt(34).

Original entry on oeis.org

5, 6, 29, 35, 379, 414, 2035, 2449, 26525, 28974, 142421, 171395, 1856371, 2027766, 9967435, 11995201, 129919445, 141914646, 697578029, 839492675, 9092504779, 9931997454, 48820494595, 58752492049
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A010489, A041057.

Programs

  • Mathematica
    Table[Numerator[FromContinuedFraction[ContinuedFraction[Sqrt[34], n]]], {n, 1, 50}] (* Vladimir Joseph Stephan Orlovsky, Mar 18 2011 *)
Numerator[Convergents[Sqrt[34],30]] (* Harvey P. Dale, Feb 17 2012 *)
CoefficientList[Series[- (x^7 - 5 x^6 + 6 x^5 - 29 x^4 - 35 x^3 - 29 x^2 - 6 x - 5)/(x^8 - 70 x^4 + 1), {x, 0, 30}], x] (* Vincenzo Librandi, Oct 23 2013 *)

Formula

a(n) = 70*a(n-4)-a(n-8). G.f.: -(x^7-5*x^6+6*x^5-29*x^4-35*x^3-29*x^2-6*x-5)/(x^8-70*x^4+1). [Colin Barker, Jul 16 2012]

A041057 Denominators of continued fraction convergents to sqrt(34).

Original entry on oeis.org

1, 1, 5, 6, 65, 71, 349, 420, 4549, 4969, 24425, 29394, 318365, 347759, 1709401, 2057160, 22281001, 24338161, 119633645, 143971806, 1559351705, 1703323511, 8372645749, 10075969260, 109132338349
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A010489, A041056.

Programs

Formula

a(n) = 70*a(n-4)-a(n-8). G.f.: -(x^2-x-1)*(x^4+6*x^2+1)/(x^8-70*x^4+1). [Colin Barker, Jul 16 2012]
Showing 1-4 of 4 results.

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

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