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A231101
a(n) = 3*a(n-3) + a(n-2), a(0)=3, a(1)=0, a(2)=2.
0
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
0,1
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.
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).
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 *)
CROSSREFS
Sequence in context: A059683 A030208 A209939 * A193084 A126598 A326602
KEYWORD
nonn,easy
AUTHOR
STATUS
approved