OFFSET
0,2
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 891
Index entries for linear recurrences with constant coefficients, signature (3,1,-2).
FORMULA
G.f.: (1-x)/(1 - 3*x - x^2 + 2*x^3)
a(n) = 3*a(n-1) + a(n-2) - 2*a(n-3).
a(n) = Sum_{alpha=RootOf(1 - 3*z - z^2 + 2*z^3)} (1/229)*(43 + 41*alpha - 46*alpha^2)*alpha^(-1-n).
a(n) = center term in M^n * [1 1 1] where M = Hosoya's triangle considered as an upper triangular 3 X 3 matrix: [2 1 2 / 1 1 0 / 1 0 0]. E.g., a(4) = 66 since M^4 * [1 1 1] = [139 66 45]. The analogous procedure using M^n * [1 0 0] generates A100058. - Gary W. Adamson, Oct 31 2004
MAPLE
spec := [S, {S=Sequence(Union(Z, Prod(Union(Sequence(Z), Z, Z), Z)))}, unlabeled]: seq(combstruct[count](spec, size=n), n=0..20);
MATHEMATICA
LinearRecurrence[{3, 1, -2}, {1, 2, 7}, 30] (* G. C. Greubel, Oct 15 2019 *)
PROG
(PARI) my(x='x+O('x^30)); Vec((1-x)/(1-3*x-x^2+2*x^3)) \\ G. C. Greubel, Oct 15 2019
(Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-x)/(1-3*x-x^2+2*x^3) )); // G. C. Greubel, Oct 15 2019
(Sage)
def A052911_list(prec):
P.<x> = PowerSeriesRing(ZZ, prec)
return P((1-x)/(1-3*x-x^2+2*x^3)).list()
A052911_list(30) # G. C. Greubel, Oct 15 2019
(GAP) a:=[1, 2, 7];; for n in [4..30] do a[n]:=3*a[n-1]+a[n-2]-2*a[n-3]; od; a; # G. C. Greubel, Oct 15 2019
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
encyclopedia(AT)pommard.inria.fr, Jan 25 2000
EXTENSIONS
More terms from James A. Sellers, Jun 06 2000
STATUS
approved