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A094170 Number of quasi-triominoes in an n X n bounding box. 5

%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 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">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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Last modified September 29 06:29 EDT 2020. Contains 337425 sequences. (Running on oeis4.)