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A242709 Nonequivalent ways to place two different markers (e.g., a pair of Go stones, black and white) on an n X n grid. 2
0, 2, 12, 33, 85, 165, 315, 518, 846, 1260, 1870, 2607, 3627, 4823, 6405, 8220, 10540, 13158, 16416, 20045, 24465, 29337, 35167, 41538, 49050, 57200, 66690, 76923, 88711, 101355, 115785, 131192, 148632, 167178, 188020, 210105, 234765, 260813, 289731, 320190 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

We say two placements are equivalent if one can be obtained from the other by rotating or reflecting the grid.

The formula was derived by categorizing and counting grid cells into four exclusive categories: central cell (if any); other diagonal cells, other horizontal and vertical midline cells (if any), and all others (in eight triangular regions) (if any); then determining for each category, how many ways a white stone could be placed in each category, given the category in which the black stone was placed prior. The sequence was verified by another program which generated all positions, removed reflections and rotations, and tallied the residue.

LINKS

Table of n, a(n) for n=1..40.

Index entries for linear recurrences with constant coefficients, signature (2,2,-6,0,6,-2,-2,1).

FORMULA

For odd n, a(n) = n*(n^3 + 3*n - 4)/8.

For even n, a(n) = n*(n^3 + n - 2)/8.

G.f.: -x^2*(x^5+x^4+7*x^3+5*x^2+8*x+2) / ((x-1)^5*(x+1)^3). - Colin Barker, May 21 2014

a(n) = n*(n^3 + n*3^(n mod 2) - 2*2^(n mod 2))/8. - Wesley Ivan Hurt, May 21 2014

MAPLE

A242709:=n->n*(n^3 + n*3^(n mod 2) - 2*2^(n mod 2))/8; seq(A242709(n), n=1..50); # Wesley Ivan Hurt, May 21 2014

MATHEMATICA

f[n_] := If[OddQ[n], (n^3 + 3 n - 4), (n^3 + n - 2)] n/8;

Table[f[n], {n, 1, 40}]

PROG

(PARI) concat(0, Vec(-x^2*(x^5+x^4+7*x^3+5*x^2+8*x+2)/((x-1)^5*(x+1)^3) + O(x^100))) \\ Colin Barker, May 21 2014

(MAGMA) [n*(n^3 + n*3^(n mod 2) - 2*2^(n mod 2))/8: n in [1..50]]; // Wesley Ivan Hurt, May 21 2014

CROSSREFS

Cf. A014409 (with indistinguishable checkers)

Sequence in context: A098203 A305330 A320542 * A055707 A055699 A244378

Adjacent sequences:  A242706 A242707 A242708 * A242710 A242711 A242712

KEYWORD

nonn,easy

AUTHOR

James Stein, May 21 2014

STATUS

approved

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Last modified July 17 08:42 EDT 2019. Contains 325098 sequences. (Running on oeis4.)