cp's OEIS Frontend

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

%I A231101 #24 May 26 2024 01:31:54
%S A231101 3,0,2,9,2,15,29,21,74,108,137,330,461,741,1451,2124,3674,6477,10046,
%T A231101 17499,29477,47637,81974,136068,224885,381990,633089,1056645,1779059,
%U A231101 2955912,4948994
%N A231101 a(n) = 3*a(n-3) + a(n-2), a(0)=3, a(1)=0, a(2)=2.
%C A231101 a(n) = r^n+s^n+t^n, where r,s,t are the roots of x^3-x-3.
%C A231101 If p is prime then p divides a(p).
%C A231101 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.
%H A231101 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (0,1,3).
%F A231101 a(n) = 3*a(n-3)+a(n-2), a(0)=3, a(1)=0, a(2)=2.
%F A231101 a(n) = r^n+s^n+t^n, where r,s,t are the roots of x^3-x-3.
%F A231101 G.f.: (x^2-3)/(3*x^3+x^2-1).
%p A231101 a:=proc(n) option remember:
%p A231101 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:
%p A231101 bign:=30:
%p A231101 seq(a(n),n=0..bign);
%t A231101 CoefficientList[Series[(x^2 - 3)/(3*x^3 + x^2 - 1), {x, 0, 50}], x] (* _Wesley Ivan Hurt_, May 26 2024 *)
%Y A231101 Cf. A001608, A072328.
%K A231101 nonn,easy
%O A231101 0,1
%A A231101 _James R. Buddenhagen_, Nov 05 2013