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%I #31 Mar 15 2024 10:11:49
%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 quasi-triominoes in an n X n bounding box.
%C A quasi-polyomino is a polyomino whose cells are not necessarily connected. For all m > 1 there are an infinite number of quasi-m-ominoes; a(n) counts the quasi-triomino (quasi-3-omino) 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 X-X
%C and
%C -X-
%C X-X
%C ---
%C whereas here they are the same quasi-polyomino.
%C a(n) can also be interpreted as the number of non-equivalent 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#order_07">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*n-3)*(-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) / ((x-1)^5*(x+1)^2). - _Colin Barker_, Feb 15 2014
%e Illustration of a(3), the 10 quasi-triominoes that fit into a 3 X 3 bounding box:
%e XXX -XX XX- X-X X-X XX- X-X X-X 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)/((x-1)^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
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