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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
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
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
KEYWORD
nonn,frac,easy
EXTENSIONS
More terms from Colin Barker, Nov 19 2013
STATUS
approved