A049673 a(n) = (F(3n) + F(n))/3, where F = A000045 (the Fibonacci sequence).

0, 1, 3, 12, 49, 205, 864, 3653, 15463, 65484, 277365, 1174889, 4976832, 21082073, 89304891, 378301260, 1602509321, 6788337557, 28755857952, 121811766781, 516002920895, 2185823443596, 9259296684333, 39223010163217, 166151337308544, 703828359351025

Offset: 0

Author

Clark Kimberling

Keywords

• nonn
• easy

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
• P. Bala, Lucas sequences and divisibility sequences
• Index entries for linear recurrences with constant coefficients, signature (5,-2,-5,-1).

Crossrefs

Cf. A000045, A000129, A001045, A215466, A273624.

Programs

Magma

[(Fibonacci(3*n)+Fibonacci(n))/3: n in [0..30]]; // Vincenzo Librandi, Jun 04 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 *)

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

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