login
This site is supported by donations to The OEIS Foundation.

 

Logo

The October issue of the Notices of the Amer. Math. Soc. has an article about the OEIS.

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A041759 Denominators of continued fraction convergents to sqrt(399). 2
1, 1, 39, 40, 1559, 1599, 62321, 63920, 2491281, 2555201, 99588919, 102144120, 3981065479, 4083209599, 159143030241, 163226239840, 6361740144161, 6524966384001, 254310462736199, 260835429120200, 10166056769303799, 10426892198423999 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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 = 38 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 27 2014

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Eric W. Weisstein, MathWorld: Lehmer Number

Index entries for linear recurrences with constant coefficients, signature (0,40,0,-1).

FORMULA

G.f.: -(x^2-x-1) / (x^4-40*x^2+1). - Colin Barker, Nov 24 2013

a(n) = 40*a(n-2) - a(n-4) for n>3, - Vincenzo Librandi, Dec 24 2013

From Peter Bala, May 27 2014: (Start)

The following remarks assume an offset of 1.

Let alpha = ( sqrt(38) + sqrt(42) )/2 and beta = ( sqrt(38) - sqrt(42) )/2 be the roots of the equation x^2 - sqrt(38)*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)} ( 38 + 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) = 38*a(2*n) + a(2*n - 1). (End)

MATHEMATICA

Denominator[Convergents[Sqrt[399], 30]] (* Vincenzo Librandi, Dec 24 2013 *)

PROG

(MAGMA) I:=[1, 1, 39, 40]; [n le 4 select I[n] else 40*Self(n-2)-Self(n-4): n in [1..40]]; // Vincenzo Librandi, Dec 24 2013

CROSSREFS

Cf. A041758, A040379. A002530.

Sequence in context: A165860 A223745 A050774 * A257442 A261374 A061756

Adjacent sequences:  A041756 A041757 A041758 * A041760 A041761 A041762

KEYWORD

nonn,frac,easy

AUTHOR

N. J. A. Sloane.

EXTENSIONS

More terms from Colin Barker, Nov 24 2013

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified September 25 22:57 EDT 2018. Contains 315425 sequences. (Running on oeis4.)