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 = 52 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 (0,54,0,-1).
FORMULA
G.f.: -(x^2-x-1) / (x^4-54*x^2+1). - Colin Barker, Dec 10 2013
a(n) = 54*a(n-2) - a(n-4) for n>3. - Vincenzo Librandi, Jan 21 2014
From Peter Bala, May 27 2014: (Start)
The following remarks assume an offset of 1.
Let alpha = sqrt(13) + sqrt(14) and beta = sqrt(13) - sqrt(14) be the roots of the equation x^2 - sqrt(52)*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)} ( 52 + 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) = 52*a(2*n) + a(2*n - 1). (End)
MATHEMATICA
Denominator/@Convergents[Sqrt[728], 40] (* Harvey P. Dale, May 08 2011 *)
CoefficientList[Series[(1 + x - x^2)/(x^4 - 54 x^2 + 1), {x, 0, 30}], x] (* Vincenzo Librandi, Jan 21 2014 *)
PROG
(Magma) I:=[1, 1, 53, 54]; [n le 4 select I[n] else 54*Self(n-2)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Jan 21 2014
CROSSREFS
KEYWORD
nonn,frac,easy
AUTHOR
EXTENSIONS
More terms from Colin Barker, Dec 10 2013
STATUS
approved