A052954 Expansion of (2-x-x^2-x^3)/((1-x)*(1-x^2-x^3)).

2, 1, 2, 2, 2, 3, 3, 4, 5, 6, 8, 10, 13, 17, 22, 29, 38, 50, 66, 87, 115, 152, 201, 266, 352, 466, 617, 817, 1082, 1433, 1898, 2514, 3330, 4411, 5843, 7740, 10253, 13582, 17992, 23834, 31573, 41825, 55406, 73397, 97230, 128802, 170626, 226031, 299427

Offset: 0

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Keywords

• easy
• nonn

Comments

For n > 2, a(n) = floor(sqrt(a(n-3)*a(n-2) + a(n-2)*a(n-1) + a(n-1)*a(n-3))). - Gerald McGarvey, Sep 19 2004

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 1025
• Dominika Závacká, Cristina Dalfó, and Miquel Angel Fiol, Integer sequences from k-iterated line digraphs, CEUR: Proc. 24th Conf. Info. Tech. - Appl. and Theory (ITAT 2024) Vol 3792, 156-161. See p. 161, Table 2.
• Index entries for linear recurrences with constant coefficients, signature (1,1,0,-1).

Crossrefs

Cf. A000931, A023434, A060006.

Programs

GAP

a:=[2,1,2,2];; for n in [5..40] do a[n]:=a[n-1]+a[n-2]-a[n-4]; od; a; # G. C. Greubel, Oct 22 2019

Magma

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( (2-x-x^2-x^3)/((1-x)*(1-x^2-x^3)) )); // G. C. Greubel, Oct 22 2019

Maple

spec:= [S,{S=Union(Sequence(Prod(Union(Prod(Z,Z),Z),Z)), Sequence(Z))}, unlabeled ]: seq(combstruct[count ](spec,size=n), n=0..20);
seq(coeff(series((2-x-x^2-x^3)/((1-x)*(1-x^2-x^3)), x, n+1), x, n), n = 0 .. 40); # G. C. Greubel, Oct 22 2019

Mathematica

LinearRecurrence[{1,1,0,-1}, {2,1,2,2}, 40] (* G. C. Greubel, Oct 22 2019 *)

PARI

my(x='x+O('x^40)); Vec((2-x-x^2-x^3)/((1-x)*(1-x^2-x^3))) \\ G. C. Greubel, Oct 22 2019

Sage

def A052954_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((2-x-x^2-x^3)/((1-x)*(1-x^2-x^3))).list()
A052954_list(40) # G. C. Greubel, Oct 22 2019

Formula

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

a(n) = a(n-2) + a(n-3) - 1.

a(n) = 1 + Sum_{alpha=RootOf(-1+z^2+z^3)} (1/23)*(3 +7*alpha -2*alpha^2) * alpha^(-1-n).

lim n->inf a(n)/a(n-1) = positive root of 1+x-x^3 (smallest Pisot-Vijayaraghavan number, A060006) - Gerald McGarvey, Sep 19 2004

a(n) = 2*A023434(n+1) - A023434(n) - A023434(n-2) - A023434(n-3). - R. J. Mathar, Nov 28 2011

a(n) = 1 + A000931(n+3). - G. C. Greubel, Oct 22 2019

Extensions

More terms from James Sellers, Jun 05 2000