OFFSET
0,1
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 910.
Index entries for linear recurrences with constant coefficients, signature (4,-4,1).
FORMULA
G.f.: (2-6*x+4*x^2-x^3)/((1-x)*(1-3*x+x^2)).
a(n) = 4*a(n-1) - 4*a(n-2) + a(n-3), with a(0)=2, a(1)=2, a(2)=4, a(3)=9.
a(n) = 1 + Sum_{alpha=RootOf(1-3*z+z^2)} (1/5)*(2-3*alpha)*alpha^(-1-n).
Product_{n>=0} (1 - 1/a(n)) = sin(Pi/10)/2. - Amiram Eldar, Jan 05 2026
MAPLE
spec:=[S, {S=Union(Sequence(Z), Sequence(Prod(Sequence(Z), Sequence(Z), Z) ))}, unlabeled]: seq(combstruct[count](spec, size=n), n=0..20);
seq(coeff(series((2-6*x+4*x^2-x^3)/((1-x)*(1-3*x+x^2)), x, n+1), x, n), n = 0..30); # G. C. Greubel, Oct 17 2019
MATHEMATICA
CoefficientList[Series[(-2+6*x-4*x^2+x^3)/(-1+x)/(1-3*x+x^2), {x, 0, 40}], x] (* Vincenzo Librandi, Jun 22 2012 *)
LinearRecurrence[{4, -4, 1}, {2, 2, 4, 9}, 30] (* G. C. Greubel, Oct 17 2019 *)
PROG
(Magma) I:=[2, 2, 4, 9]; [n le 4 select I[n] else 4*Self(n-1)-4*Self(n-2) +Self(n-3): n in [1..30]]; // Vincenzo Librandi, Jun 22 2012
(PARI) my(x='x+O('x^30)); Vec((2-6*x+4*x^2-x^3)/((1-x)*(1-3*x+x^2))) \\ G. C. Greubel, Oct 17 2019
(SageMath)
def A052925_list(prec):
P.<x> = PowerSeriesRing(ZZ, prec)
return P((2-6*x+4*x^2-x^3)/((1-x)*(1-3*x+x^2))).list()
A052925_list(30) # G. C. Greubel, Oct 17 2019
(GAP) a:=[2, 4, 9];; for n in [4..30] do a[n]:=4*a[n-1]-4*a[n-2]+a[n-3]; od; Concatenation([2], a); # G. C. Greubel, Oct 17 2019
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
INRIA Encyclopedia of Combinatorial Structures, Jan 25 2000
EXTENSIONS
More terms from James Sellers, Jun 05 2000
STATUS
approved