%I #32 Jan 25 2024 19:34:53

%S 0,3,3,1,4,4,2,5,5,3,6,6,4,7,7,5,8,8,6,9,9,7,10,10,8,11,11,9,12,12,10,

%T 13,13,11,14,14,12,15,15,13,16,16,14,17,17,15,18,18,16,19,19,17,20,20,

%U 18,21,21,19,22,22,20,23,23,21,24,24,22,25,25,23,26

%N Smallest number of integer-sided squares needed to tile a 3 X n rectangle.

%H Alois P. Heinz, <a href="/A226576/b226576.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,1,-1).

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

%F a(n) = 1 + a(n-3) for n>2; a(0)=0, a(1)=a(2)=3.

%F a(n) = (3*n+15+6*cos(2*(n-2)*Pi/3)-8*sqrt(3)*sin(2*(n-2)*Pi/3))/9. - _Wesley Ivan Hurt_, Oct 01 2017

%F a(n) = 3*floor((n+2)/3) - 2*floor(n/3). - _Ridouane Oudra_, Jan 25 2024

%e a(8) = 5:

%e ._._._._._._._._.

%e | | | |

%e | | |___|

%e |_____|_____|_|_| .

%p a:= n-> iquo(n, 3, 'r') +[0, 3, 3][r+1]:

%p seq(a(n), n=0..80);

%t CoefficientList[Series[(3 x - 2 x^3)/(1 - x - x^3 + x^4), {x, 0, 70}], x] (* _Michael De Vlieger_, Oct 01 2017 *)

%o (PARI) concat(0, Vec((3*x-2*x^3)/(1-x-x^3+x^4) + O(x^50))) \\ _Felix Fröhlich_, Oct 02 2017

%Y Cf. row m=3 of A113881, A219158.

%K nonn,easy

%O 0,2

%A _Alois P. Heinz_, Jun 12 2013