A233036 The maximum number of I-tetrominoes that can be packed into an n X n array of squares when rotation is allowed.

0, 0, 0, 4, 6, 8, 12, 16, 20, 24, 30, 36, 42, 48, 56, 64, 72, 80, 90, 100, 110, 120, 132, 144, 156, 168, 182, 196, 210, 224, 240, 256, 272, 288, 306, 324, 342, 360, 380, 400, 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

Offset: 1

Author

Kival Ngaokrajang, Dec 03 2013

Keywords

• nonn
• easy

Comments

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

Links

• N. G. de Bruijn, "Filling boxes with bricks", The American Mathematical Monthly 76 (1969), 37-40.
• Robert Israel, Illustration of initial terms
• Wikipedia, Tetromino
• Index entries for linear recurrences with constant coefficients, signature (2,-1,0,1,-2,1).

Crossrefs

Cf. A233035.

Programs

Maple

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

Mathematica

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

Formula

From Robert Israel, Oct 15 2015: (Start)

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

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

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

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

Apparently a(n) = A182568(n+2) for n > 3. - Georg Fischer, Oct 14 2018

Extensions

Corrected by Robert Israel, Oct 15 2015