OFFSET
0,3
COMMENTS
This is an odd divisibility sequence, that is, if n divides m and n/m is odd then a(n) divides a(m). More generally, if r and s are positive integers such that r = s (mod 2) then the sequence Fibonacci(r*n) + Fibonacci(s*n) is an odd divisibility sequence. In the particular case that r is even and s = r + 2 then Fibonacci(r*n) + Fibonacci(s*n) is, in fact, a divisibility sequence. See for example A215466 and A273624. - Peter Bala, May 29 2016
LINKS
Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-2,-5,-1).
FORMULA
G.f.: x*(1-2*x-x^2) / ((x^2+4*x-1)*(x^2+x-1)). - R. J. Mathar, Oct 26 2015
a(n) = 5*a(n-1) - 2*a(n-2) - 5*a(n-3) - a(n-4) for n>3. - Wesley Ivan Hurt, Jun 01 2016
a(n) = ((-(1/2*(1-sqrt(5)))^n-(2-sqrt(5))^n+(1/2*(1+sqrt(5)))^n+(2+sqrt(5))^n))/(3*sqrt(5)). - Colin Barker, Jun 02 2016
G.f.: G(F(t)), where G(t) is g.f. of A001045 and F(t) is g.f. of A000129. - Oboifeng Dira, Dec 07 2016
MAPLE
with(combinat): A049673:=n->(fibonacci(3*n)+fibonacci(n))/3: seq(A049673(n), n=0..30); # Wesley Ivan Hurt, Jun 01 2016
MATHEMATICA
Table[(Fibonacci[3 n] + Fibonacci[n])/3, {n, 0, 30}] (* Wesley Ivan Hurt, Jun 01 2016 *)
LinearRecurrence[{5, -2, -5, -1}, {0, 1, 3, 12}, 30] (* Harvey P. Dale, Sep 21 2022 *)
PROG
(PARI) concat(0, Vec(x*(1-2*x-x^2)/((x^2+4*x-1)*(x^2+x-1)) + O(x^30))) \\ Colin Barker, Jun 02 2016
(Magma) [(Fibonacci(3*n)+Fibonacci(n))/3: n in [0..30]]; // Vincenzo Librandi, Jun 04 2016
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved