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A094170 Number of quasi-triominoes in an n X n bounding box. 5
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 (list; graph; refs; listen; history; text; internal format)
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

Cf. A094171, A094172.

Sequence in context: A162433 A003012 A020478 * A004638 A211033 A020479

Adjacent sequences:  A094167 A094168 A094169 * A094171 A094172 A094173

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

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Last modified August 10 14:50 EDT 2020. Contains 336381 sequences. (Running on oeis4.)