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 1-7 of 7 results.

A248265 Egyptian fraction representation of sqrt(39) (A010493) using a greedy function.

Original entry on oeis.org

6, 5, 23, 659, 437284, 377751319913, 340271588652415528090388, 1890912187940287800367373789659912522501201614249, 7449562319978893326251035904298267810521574218546460385778180298134511070414909881921779582771096
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[ 39]]

A040032 Continued fraction for sqrt(39).

Original entry on oeis.org

6, 4, 12, 4, 12, 4, 12, 4, 12, 4, 12, 4, 12, 4, 12, 4, 12, 4, 12, 4, 12, 4, 12, 4, 12, 4, 12, 4, 12, 4, 12, 4, 12, 4, 12, 4, 12, 4, 12, 4, 12, 4, 12, 4, 12, 4, 12, 4, 12, 4, 12, 4, 12, 4, 12, 4, 12, 4, 12, 4, 12, 4, 12, 4, 12, 4
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Examples

			6.2449979983983982058468931... = 6 + 1/(4 + 1/(12 + 1/(4 + 1/(12 + ...)))). - _Harry J. Smith_, Jun 05 2009

References

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

Links

Crossrefs

Cf. A010493 (decimal expansion).

Programs

  • Maple
    Digits := 100: convert(evalf(sqrt(N)),confrac,90,'cvgts'):
  • Mathematica
    ContinuedFraction[Sqrt[39],300] (* Vladimir Joseph Stephan Orlovsky, Mar 06 2011 *)
PadRight[{6},100,{12,4}] (* or *) Join[{6},LinearRecurrence[{0,1},{4,12},100]] (* Harvey P. Dale, Feb 09 2015 *)
  • PARI
    { allocatemem(932245000); default(realprecision, 35000); x=contfrac(sqrt(39)); for (n=0, 20000, write("b040032.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 05 2009

Formula

From Stefano Spezia, Jul 27 2025: (Start)
a(n) = 2*A040008(n).
G.f.: 2*(3 + 2*x + 3*x^2)/(1 - x^2). (End)

A041064 Numerators of continued fraction convergents to sqrt(39).

Original entry on oeis.org

6, 25, 306, 1249, 15294, 62425, 764394, 3120001, 38204406, 155937625, 1909455906, 7793761249, 95434590894, 389532124825, 4769820088794, 19468812480001, 238395569848806, 973051091875225, 11915008672351506
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A041065, A010493.

Programs

Formula

G.f.: -(x^3-6*x^2-25*x-6) / (x^4-50*x^2+1). - Colin Barker, Nov 04 2013

A041065 Denominators of continued fraction convergents to sqrt(39).

Original entry on oeis.org

1, 4, 49, 200, 2449, 9996, 122401, 499600, 6117601, 24970004, 305757649, 1248000600, 15281764849, 62375059996, 763782484801, 3117504999200, 38173842475201, 155812874900004, 1907928341275249, 7787526240001000, 95358243221287249, 389220499125149996
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A010493, A041064.

Programs

  • Magma
    I:=[1, 4, 49, 200]; [n le 4 select I[n] else 50*Self(n-2)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Dec 10 2013
  • Mathematica
    Table[Denominator[FromContinuedFraction[ContinuedFraction[Sqrt[39], n]]], {n, 1, 50}] (* Vladimir Joseph Stephan Orlovsky, Mar 21 2011 *)
Denominator[Convergents[Sqrt[39], 30]] (* Vincenzo Librandi, Dec 10 2013 *)

Formula

G.f.: -(x^2-4*x-1) / (x^4-50*x^2+1). - Colin Barker, Nov 12 2013
a(n) = 50*a(n-2) - a(n-4). - Vincenzo Librandi, Dec 10 2013

Extensions

More terms from Colin Barker, Nov 12 2013

A176401 Decimal expansion of (6+sqrt(39))/2.

Original entry on oeis.org

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

Views

Author

Klaus Brockhaus, Apr 17 2010

Keywords

Comments

Continued fraction expansion of (6+sqrt(39))/2 is A010724.

Examples

			(6+sqrt(39))/2 = 6.12249899919919910292...

Links

Crossrefs

Cf. A010493 (decimal expansion of sqrt(39)), A010724 (repeat 6, 8).

A177037 Decimal expansion of (9 + 2*sqrt(39))/15.

Original entry on oeis.org

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

Views

Author

Klaus Brockhaus, May 01 2010

Keywords

Comments

Continued fraction expansion of (9 + 2*sqrt(39))/15 is A010883.
The positive solution to 15*x^2 - 18*x - 5 = 0. - Michal Paulovic, Feb 23 2023

Examples

			1.43266639978645309411...

Crossrefs

Cf. A010493 (decimal expansion of sqrt(39)), A010883 (repeat 1, 2, 3, 4).

Programs

  • Maple
    evalf(3/5 + sqrt(52/75), 100); # Michal Paulovic, Feb 24 2023
  • Mathematica
    RealDigits[(9+2*Sqrt[39])/15,10,120][[1]] (* Harvey P. Dale, Feb 12 2013 *)
  • PARI
    my(c=(9+2*quadgen(4*39))/15); a_vector(len) = digits(floor(c*10^(len-1)));
a_vector(100) \\ Kevin Ryde, Feb 24 2023

Formula

Equals sqrt(1/3 + (6/5) * sqrt(1/3 + (6/5) * sqrt(1/3 + (6/5) * ...))). - Michal Paulovic, Feb 23 2023

A176456 Decimal expansion of (12+2*sqrt(39))/3.

Original entry on oeis.org

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

Views

Author

Klaus Brockhaus, Apr 20 2010

Keywords

Comments

Continued fraction expansion of (12+2*sqrt(39))/3 is A010724 preceded by 8.

Examples

			(12+2*sqrt(39))/3 = 8.16333199893226547056...

Crossrefs

Cf. A010493 (decimal expansion of sqrt(39)), A010724 (repeat 6, 8).

Programs

  • Mathematica
    RealDigits[(12+2*Sqrt[39])/3,10,120][[1]] (* Harvey P. Dale, Mar 08 2018 *)
Showing 1-7 of 7 results.

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