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.

A041478 Numerators of continued fraction convergents to sqrt(255).

Original entry on oeis.org

15, 16, 495, 511, 15825, 16336, 505905, 522241, 16173135, 16695376, 517034415, 533729791, 16528928145, 17062657936, 528408666225, 545471324161, 16892548391055, 17438019715216, 540033139847535
Offset: 0

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Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A041479, A176110.

Programs

  • Mathematica
    Numerator[Convergents[Sqrt[255],50]] (* Harvey P. Dale, Jun 20 2012 *)
CoefficientList[Series[- (x^3 - 15 x^2 - 16 x - 15)/(x^4 - 32 x^2 + 1), {x, 0, 30}], x] (* Vincenzo Librandi, Oct 27 2013 *)

Formula

a(n) = 32*a(n-2)-a(n-4). G.f.: -(x^3-15*x^2-16*x-15)/(x^4-32*x^2+1). [Colin Barker, Jul 16 2012]

A041479 Denominators of continued fraction convergents to sqrt(255).

Original entry on oeis.org

1, 1, 31, 32, 991, 1023, 31681, 32704, 1012801, 1045505, 32377951, 33423456, 1035081631, 1068505087, 33090234241, 34158739328, 1057852414081, 1092011153409, 33818187016351, 34910198169760
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

The following remarks assume an offset of 1. This is the sequence of Lehmer numbers U_n(sqrt(R),Q) for the parameters R = 30 and Q = -1; it is a strong divisibility sequence, that is, gcd(a(n),a(m)) = a(gcd(n,m)) for all positive integers n and m. Consequently, this is a divisibility sequence: if n divides m then a(n) divides a(m). - Peter Bala, May 28 2014

Links

Crossrefs

Cf. A041478, A176110, A002530.

Programs

  • Mathematica
    Denominator[Convergents[Sqrt[255],30]] (* or *) LinearRecurrence[ {0,32,0,-1},{1,1,31,32},30] (* Harvey P. Dale, Jan 19 2013 *)
CoefficientList[Series[- (x^2 - x - 1)/(x^4 - 32 x^2 + 1), {x, 0, 30}], x] (* Vincenzo Librandi, Oct 25 2013 *)

Formula

From Colin Barker, Jul 16 2012: (Start)
a(n) = 32*a(n-2) - a(n-4).
G.f.: -(x^2-x-1)/(x^4-32*x^2+1). (End)
From Peter Bala, May 28 2014: (Start)
The following remarks assume an offset of 1.
Let alpha = ( sqrt(30) + sqrt(34) )/2 and beta = ( sqrt(30) - sqrt(34) )/2 be the roots of the equation x^2 - sqrt(30)*x - 1 = 0. Then a(n) = (alpha^n - beta^n)/(alpha - beta) for n odd, while a(n) = (alpha^n - beta^n)/(alpha^2 - beta^2) for n even.
a(n) = Product_{k = 1..floor((n-1)/2)} ( 30 + 4*cos^2(k*Pi/n) ).
Recurrence equations: a(0) = 0, a(1) = 1 and for n >= 1, a(2*n) = a(2*n - 1) + a(2*n - 2) and a(2*n + 1) = 30*a(2*n) + a(2*n - 1). (End)

A176109 Decimal expansion of (15+sqrt(255))/10.

Original entry on oeis.org

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

Views

Author

Klaus Brockhaus, Apr 10 2010

Keywords

Comments

Continued fraction expansion of (15+sqrt(255))/10 is A010708.

Examples

			(15+sqrt(255))/10 = 3.09687194226713119990...

Links

Crossrefs

Cf. A176110 (decimal expansion of sqrt(255)), A010708 (repeat 3, 10).

A176320 Decimal expansion of (15 + sqrt(255))/6.

Original entry on oeis.org

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

Views

Author

Klaus Brockhaus, Apr 15 2010

Keywords

Comments

Continued fraction expansion of (15+sqrt(255))/6 is A010717.

Examples

			(15+sqrt(255))/6 = 5.16145323711188533317...

Links

Crossrefs

Cf. A176110 (decimal expansion of sqrt(255)), A010717 (repeat 5, 6).

Programs

  • Magma
    SetDefaultRealField(RealField(120)); (15+Sqrt(255))/6; // G. C. Greubel, Nov 26 2019
  • Maple
    evalf( (15+sqrt(255))/6, 120); # G. C. Greubel, Nov 26 2019
  • Mathematica
    RealDigits[(15+Sqrt[255])/6, 10,100][[1]] (* Vincenzo Librandi, Sep 24 2013 *)
  • PARI
    default(realprecision, 120); (15+sqrt(255))/6 \\ G. C. Greubel, Nov 26 2019
  • Sage
    numerical_approx((15+sqrt(255))/6, digits=120) # G. C. Greubel, Nov 26 2019

A176397 Decimal expansion of (15+sqrt(255))/5.

Original entry on oeis.org

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

Views

Author

Klaus Brockhaus, Apr 16 2010

Keywords

Comments

Continued fraction expansion of (15+sqrt(255))/5 is A010717 preceded by 6, or equivalently A010717(n+1).

Examples

			(15+sqrt(255))/5 = 6.19374388453426239981...

Links

Crossrefs

Cf. A176110 (decimal expansion of sqrt(255)), A010717 (repeat 5, 6).

Programs

  • Mathematica
    RealDigits[(15 + Sqrt[255])/5, 10, 100][[1]]
(* Vincenzo Librandi, Sep 24 2013 *)

A176530 Decimal expansion of (15+sqrt(255))/3.

Original entry on oeis.org

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

Views

Author

Klaus Brockhaus, Apr 24 2010

Keywords

Comments

Continued fraction expansion of (15+sqrt(255))/3 is A010708 preceded by 10.

Examples

			(15+sqrt(255))/3 = 10.32290647422377066635...

Crossrefs

Cf. A176110 (decimal expansion of sqrt(255)), A010708 (repeat 3, 10).
Showing 1-6 of 6 results.

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

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