OFFSET
1,2
COMMENTS
Let P and Q be relatively prime integers. The Lucas sequence U(n) (which depends on P and Q) is an integer sequence that satisfies the recurrence equation a(n) = P*a(n-1) - Q*a(n-2) with the initial conditions U(0) = 0, U(1) = 1. The sequence {U(n)}n>=1 is a strong divisibility sequence, i.e., gcd(U(n),U(m)) = |U(gcd(n,m))|. It follows that {U(n)} is a divisibility sequence, i.e., U(n) divides U(m) whenever n divides m and U(n) <> 0.
It can be shown that if p and q are a pair of relatively prime positive integers, and if U(n) never vanishes, then the sequence {U(p*n)*U(q*n)/U(n)}n>=1 is a linear divisibility sequence of order 2*min(p,q). For a proof and a generalization of this result see the Bala link.
Here we take p = 3 and q = 4 with P = 3 and Q = 2, for which U(n) is the sequence A000225 (sometimes called the Mersenne numbers), and normalize the sequence {U(3*n)*U(4*n)/U(n)}n>=1 to have the initial term 1.
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..500
Wikipedia, Divisibility sequence
Wikipedia, Fibonacci number
Wikipedia, Lucas Sequence
Index entries for linear recurrences with constant coefficients, signature (119,-4382,59432,-280448,487424,-262144).
FORMULA
a(n) = (1/105)*(64^n + 32^n + 16^n - 4^n - 2^n - 1).
O.g.f.: x*(4096*x^4 - 4352*x^3 + 1160*x^2 - 68*x + 1 )/( (1-x)*(1-2*x)(1-4*x)*(1-16*x)*(1-32*x)*(1-64*x) ).
The formula for a(n) may be used to define it for all n in Z, and then we have a(n) = -(64)^n * a(-n). - Michael Somos, May 07 2017
EXAMPLE
G.f. = x + 51*x^2 + 2847*x^3 + 170391*x^4 + 10555655*x^5 + 664857063*x^6 + ... - Michael Somos, May 07 2017
MAPLE
seq(1/105*(2^(3*n)-1)*(2^(4*n)-1)/(2^n-1), n = 1..20);
MATHEMATICA
Table[(1/105)*(64^n + 32^n + 16^n - 4^n - 2^n - 1), {n, 1, 50}] (* G. C. Greubel, Aug 07 2018 *)
PROG
(PARI) {a(n) = if( n, (8^n - 1) * (16^n - 1) / (105 * (2^n - 1)), 0)}; /* Michael Somos, May 07 2017 */
(Magma) [(1/105)*(64^n + 32^n + 16^n - 4^n - 2^n - 1): n in [1..50]]; // G. C. Greubel, Aug 07 2018
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Peter Bala, Mar 06 2014
STATUS
approved