%I
%S 0,0,1,10,33,88,187,360,625,1024,1581,2350,3361,4680,6343,8428,10977,
%T 14080,17785,22194,27361,33400,40371,48400,57553,67968,79717,92950,
%U 107745,124264,142591,162900,185281,209920,236913,266458,298657,333720,371755,412984,457521
%N Number of quasitriominoes in an n X n bounding box.
%C 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.
%C 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:
%C 
%C X
%C XX
%C and
%C X
%C XX
%C 
%C whereas here they are the same quasipolyomino.
%C 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.
%H Vincenzo Librandi, <a href="/A094170/b094170.txt">Table of n, a(n) for n = 0..1000</a>
%H Erich Friedman, <a href="/A094170/a094170.gif">Illustration of initial terms</a>
%H <a href="/index/Rec">Index entries for linear recurrences with constant coefficients</a>, signature (3,1,5,5,1,3,1).
%F 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
%F 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
%e Illustration of a(3), the 10 quasitriominoes that fit into a 3 X 3 bounding box:
%e XXX XX XX XX XX XX XX XX X X
%e  X X X X    X X
%e      X X X X X
%t 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 *)
%o (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
%Y Cf. A094171, A094172.
%K nonn,easy
%O 0,4
%A _Jon Wild_, May 07 2004
%E Corrected and extended by _Jon Wild_, May 11 2004
%E More terms from _Colin Barker_, Feb 16 2014
