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 = 1 and Q = -1, for which U(n) is the sequence of Fibonacci numbers, A000045, and normalize the sequence 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 (14,90,-350,90,14,-1).
FORMULA
a(n) = (1/6)*Fibonacci(3*n)*Fibonacci(4*n)/Fibonacci(n).
a(n) = (1/6)*( Fibonacci(2*n) + (-1)^n*Fibonacci(4*n) + Fibonacci(6*n) ).
The sequence can be extended to negative indices when a(-n) = -a(n).
O.g.f. x*(1 + 14*x - 74*x^2 + 14*x^3 + x^4)/( (1 - 3*x + x^2)*(1 + 7*x + x^2)*(1 - 18*x + x^2) ).
Recurrence equation: a(n) = 14*a(n-1) + 90*a(n-2) - 350*a(n-3) + 90*a(n-4) + 14*a(n-5) - a(n-6).
MAPLE
with(combinat):
seq(1/6*fibonacci(3*n)*fibonacci(4*n)/fibonacci(n), n = 1..20);
MATHEMATICA
Table[(1/6)*(Fibonacci[2*n] + (-1)^n*Fibonacci[4*n] + Fibonacci[6*n]), {n, 1, 500}] (* G. C. Greubel, Aug 07 2018 *)
LinearRecurrence[{14, 90, -350, 90, 14, -1}, {1, 28, 408, 7896, 137555, 2496144}, 20] (* Harvey P. Dale, Aug 26 2020 *)
PROG
(PARI) vector(30, n, (fibonacci(2*n) + (-1)^n*fibonacci(4*n) + fibonacci(6*n))/6) \\ G. C. Greubel, Aug 07 2018
(Magma) [(Fibonacci(2*n) + (-1)^n*Fibonacci(4*n) + Fibonacci(6*n))/6: n in [1..30]]; // G. C. Greubel, Aug 07 2018
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Peter Bala, Mar 01 2014
STATUS
approved