A042199 Denominators of continued fraction convergents to sqrt(624).

1, 1, 49, 50, 2449, 2499, 122401, 124900, 6117601, 6242501, 305757649, 312000150, 15281764849, 15593764999, 763782484801, 779376249800, 38173842475201, 38953218725001, 1907928341275249, 1946881560000250, 95358243221287249, 97305124781287499

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 = 48 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..100

Eric W. Weisstein, MathWorld: Lehmer Number

Index entries for linear recurrences with constant coefficients, signature (0,50,0,-1).

Formula

G.f.: -(x^2-x-1) / (x^4-50*x^2+1). - Colin Barker, Nov 19 2013

From Peter Bala, May 27 2014: (Start)

The following remarks assume an offset of 1.

Let alpha = sqrt(12) + sqrt(13) and beta = sqrt(12) - sqrt(13) be the roots of the equation x^2 - sqrt(48)*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)} ( 48 + 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) = 48*a(2*n) + a(2*n - 1). (End)

Mathematica

Denominator[Convergents[Sqrt[624], 30]] (* Harvey P. Dale, Sep 22 2013 *)

Crossrefs

Cf. A042198, A040599. A002530.

Sequence in context: A037412 A176309 A080201 * A257443 A020276 A346805

Adjacent sequences: A042196 A042197 A042198 * A042200 A042201 A042202

Keyword

nonn,frac,easy

Author

N. J. A. Sloane

Extensions

More terms from Colin Barker, Nov 19 2013

Status

approved