cp's OEIS Frontend

A060312 Number of distinct ways to tile a 2 X n rectangle with dominoes (solutions are identified if they are rotations or reflections of each other).

%I A060312 #41 Aug 26 2025 08:38:28
%S A060312 1,1,2,4,5,9,12,21,30,51,76,127,195,322,504,826,1309,2135,3410,5545,
%T A060312 8900,14445,23256,37701,60813,98514,159094,257608,416325,673933,
%U A060312 1089648,1763581,2852242,4615823,7466468,12082291,19546175,31628466
%N A060312 Number of distinct ways to tile a 2 X n rectangle with dominoes (solutions are identified if they are rotations or reflections of each other).
%C A060312 Same as A001224 except that there a(2)=2 not 1. - _N. J. A. Sloane_, Mar 30 2015
%H A060312 G. C. Greubel, <a href="/A060312/b060312.txt">Table of n, a(n) for n = 1..1001</a>
%H A060312 A. R. Ashrafi, J. Azarija, K. Fathalikhani, S. Klavzar, et al., <a href="http://www.fmf.uni-lj.si/~klavzar/preprints/Fib-Luc-orbits-August-11-2014.pdf">Orbits of Fibonacci and Lucas cubes, dihedral transformations, and asymmetric strings</a>, 2014.
%H A060312 R. J. Mathar, <a href="http://arxiv.org/abs/1311.6135">Paving rectangular regions with rectangular tiles, ...</a>, arXiv:1311.6135 [math.CO], Table 9.
%H A060312 W. E. Patten (proposer) and S. W. Golomb (solver), <a href="http://www.jstor.org/stable/2312751">Problem E1470</a>, "Covering a 2Xn rectangle with dominoes", Amer. Math. Monthly, 69 (1962), 61-62.
%H A060312 N. J. A. Sloane, <a href="/A001224/a001224.png">Annotated scan of Monthly problem E1470 with illustration of a(4)=4 (Page 1)</a>
%H A060312 N. J. A. Sloane, <a href="/A001224/a001224_1.png">Annotated scan of Monthly problem E1470 with illustration of a(4)=4 (Page 2)</a>
%H A060312 <a href="/index/Do#domino">Index entries for sequences related to dominoes</a>
%H A060312 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-1,0,-1,-1).
%F A060312 If F(n) is the n-th Fibonacci number, then a(2n) = (F(2n+1) + F(n+2))/2 for n > 1 and a(2n-1) = (F(2n) + F(n))/2. [Corrected by _Manfred Boergens_, Aug 25 2025]
%F A060312 a(n) = (F(n+1)+F(floor((n+3+(-1)^n)/2)))/2 for n!=2. - _Manfred Boergens_, Aug 25 2025
%F A060312 G.f.: -x*(x^7 + x^6 + x^5 + 2*x^4 - x^3 + x^2 - 1) / ((x^2 + x - 1)*(x^4 + x^2 - 1)). - _Colin Barker_, Dec 15 2012
%e A060312 a(3)=2 because of the configurations |= and |||.
%p A060312 # Maple code for A060312 and A001224 from _N. J. A. Sloane_, Mar 30 2015
%p A060312 with(combinat); F:=fibonacci;
%p A060312 f:=proc(n) option remember;
%p A060312 if n=2 then 1 # change this to 2 to get A001224
%p A060312 elif (n mod 2) = 0 then (F(n+1)+F(n/2+2))/2;
%p A060312 else (F(n+1)+F((n+1)/2))/2; fi; end;
%p A060312 [seq(f(n),n=1..50)];
%t A060312 CoefficientList[Series[-(x^7 + x^6 + x^5 + 2 x^4 - x^3 + x^2 - 1) / ((x^2 + x - 1) (x^4 + x^2 - 1)), {x, 0, 40}], x] (* _Vincenzo Librandi_, Nov 22 2014 *)
%o A060312 (Magma) [n eq 1 select 1 else (1/2)*(Fibonacci(n+2)+Fibonacci(Floor((n-(-1)^n)/2)+2)): n in [0..40]]; // _Vincenzo Librandi_, Nov 22 2014
%Y A060312 Essentially the same as A001224, which is the main entry for this sequence. Other versions of the sequence can be found in A068928 and A102526.
%K A060312 easy,nonn,changed
%O A060312 1,3
%A A060312 _Thomas Ward_, Mar 27 2001
%E A060312 Edited by _N. J. A. Sloane_, Mar 30 2015