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

%I A253265 #27 Sep 08 2022 08:46:10
%S A253265 1,7,82,877,9565,103960,1130701,12296275,133724242,1454268793,
%T A253265 15815379409,171994465072,1870463946217,20341557798991,
%U A253265 221217294787570,2405769114915733,26163076626035413,284527128680078536,3094272440210485525,33650646877362841531,365955505581792121138
%N A253265 The number of tilings of 2 X n boards with squares of 2 colors and dominoes of 3 colors.
%C A253265 The numerator in Formula (3) in the JIS article should be 1-b*x, not 1-x.
%H A253265 G. C. Greubel, <a href="/A253265/b253265.txt">Table of n, a(n) for n = 0..950</a>
%H A253265 M. Katz, C. Stenson, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/Stenson/stenson8.html">Tiling a (2 x n)-board with squares and dominoes</a>, JIS 12 (2009) 09.2.2, Table 1, a=2, b=3.
%H A253265 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (10,12,-27).
%F A253265 G.f.: ( 1-3*x ) / ( 1 - 10*x - 12*x^2 + 27*x^3 ).
%p A253265 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
%t A253265 CoefficientList[Series[(1-3x)/(1-10x-12x^2+27x^3), {x, 0, 20}], x] (* _Michael De Vlieger_, Sep 30 2015 *)
%t A253265 LinearRecurrence[{10,12,-27},{1,7,82},30] (* _Harvey P. Dale_, Dec 30 2015 *)
%o A253265 (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
%o A253265 (Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-3*x)/(1-10*x-12*x^2+27*x^3) )); // _G. C. Greubel_, Oct 28 2019
%o A253265 (Sage)
%o A253265 def A253265_list(prec):
%o A253265     P.<x> = PowerSeriesRing(ZZ, prec)
%o A253265     return P((1-3*x)/(1-10*x-12*x^2+27*x^3)).list()
%o A253265 A253265_list(30) # _G. C. Greubel_, Oct 28 2019
%o A253265 (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
%Y A253265 Cf. A030186 (pieces of a single color), A102436.
%K A253265 nonn,easy
%O A253265 0,2
%A A253265 _R. J. Mathar_, Sep 30 2015