

A094170


Number of quasitriominoes in an n X n bounding box.


5



0, 0, 1, 10, 33, 88, 187, 360, 625, 1024, 1581, 2350, 3361, 4680, 6343, 8428, 10977, 14080, 17785, 22194, 27361, 33400, 40371, 48400, 57553, 67968, 79717, 92950, 107745, 124264, 142591, 162900, 185281, 209920, 236913, 266458, 298657, 333720, 371755, 412984, 457521
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OFFSET

0,4


COMMENTS

A quasipolyomino is a polyomino whose cells are not necessarily connected. For all m > 1 there are an infinite number of quasimominoes; a(n) counts the quasitriomino (quasi3omino) equivalence classes (under translation, rotation by 90 degrees and vertical and horizontal symmetry) whose members fit into an n X n bounding box.
This is different from A082966 because that sequence considers these two (for example) as different ways of placing 3 counters on a 3 X 3 checkerboard:

X
XX
and
X
XX

whereas here they are the same quasipolyomino.
a(n) can also be interpreted as the number of nonequivalent Game of Life patterns on an n X n board that have exactly 3 live cells, etc.


LINKS



FORMULA

a(n) = (1/32)*(6*n^4  12*n^3 + 32*n^2  58*n + 29  (6*n3)*(1)^n).  Ralf Stephan, Dec 03 2004
G.f.: x^2*(x^5+x^4+4*x^3+4*x^2+7*x+1) / ((x1)^5*(x+1)^2).  Colin Barker, Feb 15 2014


EXAMPLE

Illustration of a(3), the 10 quasitriominoes that fit into a 3 X 3 bounding box:
XXX XX XX XX XX XX XX XX X X
 X X X X    X X
     X X X X X


MATHEMATICA

CoefficientList[Series[x^2 (x^5 + x^4 + 4 x^3 + 4 x^2 + 7 x + 1)/((1  x)^5 (x + 1)^2), {x, 0, 50}], x] (* Vincenzo Librandi, Feb 17 2014 *)


PROG

(PARI) Vec(x^2*(x^5+x^4+4*x^3+4*x^2+7*x+1)/((x1)^5*(x+1)^2) + O(x^100)) \\ Colin Barker, Feb 16 2014


CROSSREFS



KEYWORD

nonn,easy


AUTHOR



EXTENSIONS

Corrected and extended by Jon Wild, May 11 2004


STATUS

approved



