A233036 - OEIS

%I #65 Jun 07 2019 07:48:22

%S 0,0,0,4,6,8,12,16,20,24,30,36,42,48,56,64,72,80,90,100,110,120,132,

%T 144,156,168,182,196,210,224,240,256,272,288,306,324,342,360,380,400,

%U 420,440,462,484,506,528,552,576,600,624,650,676,702,728,756,784,812,840,870,900,930,960,992,1024,1056

%N The maximum number of I-tetrominoes that can be packed into an n X n array of squares when rotation is allowed.

%C By de Bruijn's theorem (see the de Bruijn link), an m X n rectangle can't be tiled with I tetrominoes unless m or n is divisible by 4. - _Robert Israel_, Oct 15 2015

%H N. G. de Bruijn, <a href="http://dx.doi.org/10.2307/2316785">"Filling boxes with bricks"</a>, The American Mathematical Monthly 76 (1969), 37-40.

%H Robert Israel, <a href="/A233036/a233036_1.pdf">Illustration of initial terms</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Tetromino">Tetromino</a>

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

%F From _Robert Israel_, Oct 15 2015: (Start)

%F a(4*k) = 4*k^2.

%F a(2*k+1) = k*(k+1) for k >= 2.

%F a(4*k+2) = 4*k*(k+1).

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

%F Apparently a(n) = A182568(n+2) for n > 3. - _Georg Fischer_, Oct 14 2018

%p 0$3, seq(op([4*k^2, 2*k*(2*k+1),4*k*(k+1),(2*k+1)*(2*k+2)]),k=1..20);# _Robert Israel_, Oct 15 2015

%t CoefficientList[Series[2 x^3/((1 + x) (1 + x^2) (1 - x)^3) - 2 x^3, {x, 0, 100}], x] (* _Vincenzo Librandi_, Oct 15 2015 *)

%t LinearRecurrence[{2,-1,0,1,-2,1},{0,0,0,4,6,8,12,16,20},70] (* _Harvey P. Dale_, Dec 16 2018 *)

%Y Cf. A233035.

%K nonn,easy

%O 1,4

%A _Kival Ngaokrajang_, Dec 03 2013

%E Corrected by _Robert Israel_, Oct 15 2015