A231101 a(n) = 3*a(n-3) + a(n-2), a(0)=3, a(1)=0, a(2)=2.

3, 0, 2, 9, 2, 15, 29, 21, 74, 108, 137, 330, 461, 741, 1451, 2124, 3674, 6477, 10046, 17499, 29477, 47637, 81974, 136068, 224885, 381990, 633089, 1056645, 1779059, 2955912, 4948994

Offset: 0

Author

James R. Buddenhagen, Nov 05 2013

Keywords

• nonn
• easy

Comments

a(n) = r^n+s^n+t^n, where r,s,t are the roots of x^3-x-3.

If p is prime then p divides a(p).

Both this and the Perrin sequence are linear recurrences with a(n) depending on a(n-3) and a(n-2) but not on a(n-1), with the same initial conditions; both are sums of powers of roots of a cubic: Perrin: a(n) = r^n+s^n+t^n with r,s,t roots of x^3-x-1 this seq: a(n) = r^n+s^n+t^n with r,s,t roots of x^3-x-3. See crossrefs.

Links

• Index entries for linear recurrences with constant coefficients, signature (0,1,3).

Crossrefs

Cf. A001608, A072328.

Programs

Maple

a:=proc(n) option remember:
if n=0 then 3 elif n=1 then 0 elif n=2 then 2 else 3*a(n-3)+a(n-2) end if end proc:
bign:=30:
seq(a(n),n=0..bign);

Mathematica

CoefficientList[Series[(x^2 - 3)/(3*x^3 + x^2 - 1), {x, 0, 50}], x] (* Wesley Ivan Hurt, May 26 2024 *)

Formula

a(n) = 3*a(n-3)+a(n-2), a(0)=3, a(1)=0, a(2)=2.

a(n) = r^n+s^n+t^n, where r,s,t are the roots of x^3-x-3.

G.f.: (x^2-3)/(3*x^3+x^2-1).