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 = 58 and Q = -1. This 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 26 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 (0,60,0,-1).
FORMULA
G.f.: -(x^2-x-1) / (x^4-60*x^2+1). - Colin Barker, Dec 22 2013
From Peter Bala, May 26 2014: (Start)
The following remarks assume an offset of 1. Let alpha = ( sqrt(58) + sqrt(62) )/2 and beta = ( sqrt(58) - sqrt(62) )/2 be the roots of the equation x^2 - sqrt(58)*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)} (58 + 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) = 58*a(2*n) + a(2*n - 1). (End)
MATHEMATICA
Denominator[Convergents[Sqrt[899], 30]] (* Vincenzo Librandi, Jan 28 2014 *)
LinearRecurrence[{0, 60, 0, -1}, {1, 1, 59, 60}, 30] (* Harvey P. Dale, Apr 01 2017 *)
CROSSREFS
KEYWORD
nonn,frac,easy,changed
AUTHOR
EXTENSIONS
Additional term from Colin Barker, Dec 22 2013
STATUS
approved