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A094170
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Number of quasi-triominoes in an n X n bounding box.
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5
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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
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0,4
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COMMENTS
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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.
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-
X-X
and
-X-
X-X
---
whereas here they are the same quasi-polyomino.
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.
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LINKS
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FORMULA
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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
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
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EXAMPLE
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Illustration of a(3), the 10 quasi-triominoes that fit into a 3 X 3 bounding box:
XXX -XX XX- X-X X-X XX- X-X X-X X-- X--
--- -X- --X X-- -X- --- --- --- -X- --X
--- --- --- --- --- --X X-- -X- --X -X-
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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Corrected and extended by Jon Wild, May 11 2004
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STATUS
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approved
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