A278815 - OEIS

%I #30 Sep 08 2022 08:46:18

%S 1,2,7,29,109,416,1596,6105,23362,89415,342193,1309593,5011920,

%T 19180976,73406985,280933906,1075154535,4114694797,15747237101,

%U 60265824784,230641706484,882682631025,3378090801226,12928199853783,49477163668857,189352713633433

%N Number of tilings of a 2 X n grid with monomers, dimers, and trimers.

%C The first three terms are the same as A030186 because there are only monomers and dimers in boards with n<3.

%H Alois P. Heinz, <a href="/A278815/b278815.txt">Table of n, a(n) for n = 0..1000</a>

%H Kathryn Haymaker and Sara Robertson, <a href="https://www.emis.de/journals/JIS/VOL20/Haymaker/haym2.html">Counting Colorful Tilings of Rectangular Arrays</a>, Journal of Integer Sequences, Vol. 20 (2017), Article 17.5.8, Corollary 2.

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (3,2,5,-2,0,-1).

%F a(n) = 3*a(n-1) + 2*a(n-2) + 5*a(n-3) - 2*a(n-4) - a(n-6).

%F G.f.: (1 - x - x^2 - x^3)/(1 - 3*x - 2*x^2 - 5*x^3 + 2*x^4 + x^6).

%p seq(coeff(series((1-x-x^2-x^3)/(1-3*x-2*x^2-5*x^3+2*x^4+x^6), x, n+1), x, n), n = 0..30); # _G. C. Greubel_, Oct 28 2019

%t LinearRecurrence[{3,2,5,-2,0,-1}, {1,2,7,29,109,416}, 30] (* _G. C. Greubel_, Oct 28 2019 *)

%o (PARI) my(x='x+O('x^30)); Vec((1-x-x^2-x^3)/(1-3*x-2*x^2-5*x^3+ 2*x^4 +x^6)) \\ _G. C. Greubel_, Oct 28 2019

%o (Magma) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-x-x^2-x^3)/(1-3*x-2*x^2-5*x^3+2*x^4+x^6) )); // _G. C. Greubel_, Oct 28 2019

%o (Sage)

%o def A278815_list(prec):

%o     P.<x> = PowerSeriesRing(ZZ, prec)

%o     return P( (1-x-x^2-x^3)/(1-3*x-2*x^2-5*x^3+2*x^4+x^6) ).list()

%o A278815_list(30) # _G. C. Greubel_, Oct 28 2019

%o (GAP) a:=[1,2,7,29,109,416];; for n in [7..30] do a[n]:=3*a[n-1]+2*a[n-2] +5*a[n-3]-2*a[n-4]-a[n-6]; od; a; # _G. C. Greubel_, Oct 28 2019

%Y Cf. A030186, A052961, A129682, A219866.

%K nonn,easy

%O 0,2

%A _Kathryn Haymaker_, Nov 28 2016