|
%I #45 Jun 26 2022 10:47:46
%S 1,3,10,32,103,331,1064,3420,10993,35335,113578,365076,1173471,
%T 3771911,12124128,38970824,125264689,402640763,1294216154,4160024536,
%U 13371648999,42980755379,138153890600,444070778180,1427385469761,4588073296863,14747534582170
%N Expansion of 1/(1 - 3*x - x^2 + x^3).
%C From _Greg Dresden_, Jun 25 2022: (Start)
%C a(n) is the number of ways to tile, with squares and dominoes, a 2 X n board with one extra space at the end. Here is the board for n=3:
%C _____
%C |_|_|_|_
%C |_|_|_|_|
%C and here is one of the a(3)=32 possible tilings of this board:
%C _____
%C | |_|_|_
%C |_|_|___|
%C (End)
%H Vincenzo Librandi, <a href="/A033505/b033505.txt">Table of n, a(n) for n = 0..1000</a>
%H Richard M. Low and Ardak Kapbasov, <a href="https://www.emis.de/journals/JIS/VOL20/Low/low2.html">Non-Attacking Bishop and King Positions on Regular and Cylindrical Chessboards</a>, Journal of Integer Sequences, Vol. 20 (2017), Article 17.6.1, Table 4.
%H N. J. A. Sloane <a href="/A030186/a030186.txt">Notes on A030186 and A033505</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,1,-1).
%F a(n) = 3*a(n-1) + a(n-2) - a(n-3). - _Greg Dresden_, Aug 16 2018
%p seq(coeff(series(1/(1-3*x-x^2+x^3), x, n+1), x, n), n = 0..30); # _G. C. Greubel_, Oct 14 2019
%t CoefficientList[Series[1/(1-3x-x^2+x^3), {x, 0, 30}], x] (* or *) LinearRecurrence[{3,1,-1}, {1,3,10}, 30] (* _Vincenzo Librandi_, Aug 17 2018 *)
%o (Magma) I:=[1,3,10]; [n le 3 select I[n] else 3*Self(n-1)+Self(n-2)-Self(n-3): n in [1..30]]; // _Vincenzo Librandi_, Aug 17 2018
%o (PARI) my(x='x+O('x^30)); Vec(1/(1-3*x-x^2+x^3)) \\ _G. C. Greubel_, Oct 14 2019
%o (Sage)
%o def A033505_list(prec):
%o P.<x> = PowerSeriesRing(ZZ, prec)
%o return P(1/(1-3*x-x^2+x^3)).list()
%o A033505_list(30) # _G. C. Greubel_, Oct 14 2019
%o (GAP) a:=[1,3,10];; for n in [4..30] do a[n]:=3*a[n-1]+a[n-2]-a[n-3]; od; a; # _G. C. Greubel_, Oct 14 2019
%Y Partial sums of A030186.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_, Feb 13 2002
|
