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

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A041179 Denominators of continued fraction convergents to sqrt(99). 2
1, 1, 19, 20, 379, 399, 7561, 7960, 150841, 158801, 3009259, 3168060, 60034339, 63202399, 1197677521, 1260879920, 23893516081, 25154396001, 476672644099, 501827040100, 9509559365899, 10011386405999, 189714514673881, 199725901079880, 3784780734111721 (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 = 18 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

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 (0,20,0,-1).

FORMULA

G.f.: -(x^2-x-1) / (x^4-20*x^2+1). - Colin Barker, Nov 14 2013

a(n) = 20*a(n-2) - a(n-4). - Vincenzo Librandi, Dec 12 2013

From Peter Bala, May 28 2014: (Start)

The following remarks assume an offset of 1.

Let alpha = ( sqrt(18) + sqrt(22) )/2 and beta = ( sqrt(18) - sqrt(22) )/2 be the roots of the equation x^2 - sqrt(18)*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)} ( 18 + 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) = 18*a(2*n) + a(2*n - 1). (End)

MATHEMATICA

Denominator[Convergents[Sqrt[99], 30]] (* Vincenzo Librandi, Dec 12 2013 *)

PROG

(MAGMA) I:=[1, 1, 19, 20]; [n le 4 select I[n] else 20*Self(n-2)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Dec 12 2013

CROSSREFS

Cf. A041178, A010170, A020856, A010550, A002530.

Sequence in context: A041752 A041754 A197353 * A041756 A041757 A041758

Adjacent sequences:  A041176 A041177 A041178 * A041180 A041181 A041182

KEYWORD

nonn,frac,easy

AUTHOR

N. J. A. Sloane.

EXTENSIONS

More terms from Colin Barker, Nov 14 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
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified November 21 06:00 EST 2019. Contains 329350 sequences. (Running on oeis4.)