A253265 The number of tilings of 2 X n boards with squares of 2 colors and dominoes of 3 colors.

1, 7, 82, 877, 9565, 103960, 1130701, 12296275, 133724242, 1454268793, 15815379409, 171994465072, 1870463946217, 20341557798991, 221217294787570, 2405769114915733, 26163076626035413, 284527128680078536, 3094272440210485525, 33650646877362841531, 365955505581792121138

Offset: 0

Author

R. J. Mathar, Sep 30 2015

Keywords

• nonn
• easy

Comments

The numerator in Formula (3) in the JIS article should be 1-b*x, not 1-x.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..950
• M. Katz, C. Stenson, Tiling a (2 x n)-board with squares and dominoes, JIS 12 (2009) 09.2.2, Table 1, a=2, b=3.
• Index entries for linear recurrences with constant coefficients, signature (10,12,-27).

Crossrefs

Cf. A030186 (pieces of a single color), A102436.

Programs

GAP

a:=[1,7,82];; for n in [4..30] do a[n]:=10*a[n-1]+12*a[n-2] -27*a[n-3]; od; a; # G. C. Greubel, Oct 28 2019

Magma

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-3*x)/(1-10*x-12*x^2+27*x^3) )); // G. C. Greubel, Oct 28 2019

Maple

seq(coeff(series((1-3*x)/(1-10*x-12*x^2+27*x^3), x, n+1), x, n), n = 0..30); # G. C. Greubel, Oct 28 2019

Mathematica

CoefficientList[Series[(1-3x)/(1-10x-12x^2+27x^3), {x, 0, 20}], x] (* Michael De Vlieger, Sep 30 2015 *)
LinearRecurrence[{10,12,-27},{1,7,82},30] (* Harvey P. Dale, Dec 30 2015 *)

PARI

my(x='x+O('x^30)); Vec((1-3*x)/(1-10*x-12*x^2+27*x^3)) \\ G. C. Greubel, Oct 28 2019

Sage

def A253265_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P((1-3*x)/(1-10*x-12*x^2+27*x^3)).list()
A253265_list(30) # G. C. Greubel, Oct 28 2019

Formula

G.f.: ( 1-3*x ) / ( 1 - 10*x - 12*x^2 + 27*x^3 ).