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 = 6 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 28 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 (0,8,0,-1).
FORMULA
G.f.: (1+x-x^2)/(1-8*x^2+x^4). - Colin Barker, Jan 01 2012
From Peter Bala, May 28 2014: (Start)
The following remarks assume an offset of 1.
Let alpha = ( sqrt(6) + sqrt(10) )/2 and beta = ( sqrt(6) - sqrt(10) )/2 be the roots of the equation x^2 - sqrt(6)*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)} ( 6 + 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) = 6*a(2*n) + a(2*n - 1). (End)
From Gerry Martens, Jul 11 2015: (Start)
Interspersion of 2 sequences [a0(n),a1(n)] for n>0:
a0(n) = (-((-5+sqrt(15))*(4+sqrt(15))^n)+(4-sqrt(15))^n*(5+sqrt(15)))/10.
a1(n) = (-(4-sqrt(15))^n+(4+sqrt(15))^n)/(2*sqrt(15)). (End)
MATHEMATICA
Denominator[NestList[(6/(6+#))&, 0, 60]] (* Vladimir Joseph Stephan Orlovsky, Apr 13 2010 *)
a0[n_] := (-((-5+Sqrt[15])*(4+Sqrt[15])^n)+(4-Sqrt[15])^n*(5+Sqrt[15]))/10 // Simplify
a1[n_] := (-(4-Sqrt[15])^n+(4+Sqrt[15])^n)/(2*Sqrt[15]) // Simplify
Flatten[MapIndexed[{a0[#], a1[#]} &, Range[20]]] (* Gerry Martens, Jul 11 2015 *)
Convergents[Sqrt[15], 30]//Denominator (* Harvey P. Dale, Aug 13 2016 *)
CROSSREFS
KEYWORD
nonn,cofr,frac,easy
AUTHOR
STATUS
approved