A121907 Expansion of g.f.: (1 + x + x^2)/(1 - 2*x - 2*x^2).

1, 3, 9, 24, 66, 180, 492, 1344, 3672, 10032, 27408, 74880, 204576, 558912, 1526976, 4171776, 11397504, 31138560, 85072128, 232421376, 634987008, 1734816768, 4739607552, 12948848640, 35376912384, 96651522048, 264056868864, 721416781824, 1970947301376

Offset: 0

Author

N. J. A. Sloane, Nov 20 2006

Keywords

• nonn
• easy

Comments

a(n-1) is the number of compositions of n into floor((3*j-1)/2) kinds of j's for all j >= 1. The sequence of such compositions is 1,1,3,9,24,... (i.e., this sequence prepended by 1) and has g.f. 1/(1-Sum_{j>=1} floor((3*j-1)/2)*x^j). - Joerg Arndt, Jul 06 2011

a(n) is the number of length n words on 3 letters (ternary words) such that the length of any run of identical letters is <= 2. Cf. A181137 for a generalization. - Geoffrey Critzer, Sep 16 2013

References

• A. Burstein and T. Mansour, Words restricted by 3-letter generalized multipermutation patterns, Annals. Combin., 7 (2003), 1-14.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• A. Burstein and T. Mansour, Words restricted by 3-letter generalized multipermutation patterns, arXiv:math/0112281 [math.CO], 2001.
• Martin Burtscher, Igor Szczyrba, Rafał Szczyrba, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
• Index entries for linear recurrences with constant coefficients, signature (2,2).

Crossrefs

Column 3 in A265584.

Programs

GAP

a:=[3,9];; for n in [3..30] do a[n]:=2*(a[n-1]+a[n-2]); od; Concatenation([1], a); # G. C. Greubel, Oct 07 2019

Magma

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1+x+x^2)/(1-2*x-2*x^2) )); // G. C. Greubel, Oct 07 2019

Maple

seq(coeff(series((1+x+x^2)/(1-2*x-2*x^2), x, n+1), x, n), n = 0..30); # G. C. Greubel, Oct 07 2019

Mathematica

CoefficientList[Series[(1+x+x^2)/(1-2x-2x^2),{x,0,30}],x] (* or *) LinearRecurrence[{2,2},{1,3,9},30] (* Harvey P. Dale, Dec 03 2011 *)

PARI

my(x='x+O('x^30)); Vec((1+x+x^2)/(1-2*x-2*x^2)) \\ G. C. Greubel, Oct 07 2019

Sage

def A121907_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1+x+x^2)/(1-2*x-2*x^2)).list()
A121907_list(30) # G. C. Greubel, Oct 07 2019

Formula

a(0)=1, a(1)=3, a(2)=9; a(n) = 2*a(n-1) + 2*a(n-2) for n>=3. - Philippe Deléham, Sep 19 2009

G.f.: (1/x)*(-1 + 1/(1-Sum_{j>=1} floor((3*j-1)/2)*x^j)). - Joerg Arndt, Jul 06 2011

E.g.f.: (1/2)*exp(x)*(3*cosh(sqrt(3)*x) + sqrt(3)*sinh(sqrt(3)*x)) - 1/2. - Stefano Spezia, Oct 08 2019