

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

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Erich Friedman, Illustration of initial terms
Index entries for linear recurrences with constant coefficients, signature (3,1,5,5,1,3,1).


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

Cf. A094171, A094172.
Sequence in context: A162433 A003012 A020478 * A004638 A211033 A020479
Adjacent sequences: A094167 A094168 A094169 * A094171 A094172 A094173


KEYWORD

nonn,easy


AUTHOR

Jon Wild, May 07 2004


EXTENSIONS

Corrected and extended by Jon Wild, May 11 2004
More terms from Colin Barker, Feb 16 2014


STATUS

approved



