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

1,4

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

Table of n, a(n) for n=1..65.

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).

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

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 *)

Crossrefs

Cf. A233035.

Sequence in context: A076082 A162648 A225512 * A062554 A020225 A310663

Adjacent sequences: A233033 A233034 A233035 * A233037 A233038 A233039

Keyword

nonn,easy

Author

Kival Ngaokrajang, Dec 03 2013

Extensions

Corrected by Robert Israel, Oct 15 2015

Status

approved