OFFSET
0,4
COMMENTS
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.
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*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
EXAMPLE
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-
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)/((x-1)^5*(x+1)^2) + O(x^100)) \\ Colin Barker, Feb 16 2014
CROSSREFS
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