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A042199 Denominators of continued fraction convergents to sqrt(624). 2
1, 1, 49, 50, 2449, 2499, 122401, 124900, 6117601, 6242501, 305757649, 312000150, 15281764849, 15593764999, 763782484801, 779376249800, 38173842475201, 38953218725001, 1907928341275249, 1946881560000250, 95358243221287249, 97305124781287499 (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 = 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 A118073

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

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Last modified January 18 08:37 EST 2021. Contains 340250 sequences. (Running on oeis4.)