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A232568 Number of non-equivalent binary n X n matrices with three pairwise nonadjacent 1's. 4
0, 6, 40, 210, 681, 1919, 4443, 9481, 18206, 33164, 56570, 92996, 146175, 223565, 330981, 479779, 678508, 943586, 1287036, 1731654, 2293765, 3004011, 3883935, 4973645, 6300906, 7917064, 9857198, 12185816, 14946491, 18218969, 22056585, 26556551, 31783320 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,2

COMMENTS

Also: Number of non-equivalent ways to place three non-attacking wazirs on an n X n board.

Two matrix elements are considered adjacent if the difference of their row indices is 1 and the column indices are equal, or vice versa (von Neumann neighborhood).

Counted for this sequence are equivalence classes induced by the dihedral group D_4. If equivalent matrices are being destinguished, the number of matrices is A172226(n).

LINKS

Heinrich Ludwig, Table of n, a(n) for n = 2..1001

Index entries for linear recurrences with constant coefficients, signature (3,1,-11,6,14,-14,-6,11,-1,-3,1).

FORMULA

a(n) = (n^6 - 15*n^4 + 20*n^3 + 50*n^2 - 116*n + 48)/48 if n is even; a(n) = (n^6 - 15*n^4 + 28*n^3 + 29*n^2 - 76*n - 15)/48 if n is odd.

G.f.: x^3*(x^9-4*x^8+x^7+12*x^6+9*x^5-70*x^4-77*x^3-84*x^2-22*x-6) / ((x-1)^7*(x+1)^4). - Colin Barker, Dec 06 2013

a(n) = (n^6 - 15n^4 + 28n^3 + 29n^2 - 76n - 15 - ((n+1) mod 2) * (8n^3 - 21n^2 + 40n - 63))/48. - Wesley Ivan Hurt, Dec 06 2013

EXAMPLE

There are a(3) = 6 non-equivalent 3 X 3 matrices with three pairwise nonadjacent 1's (and no other 1's):

  [1 0 0]    [1 0 1]    [1 0 0]    [1 0 1]   [1 0 1]   [0 1 0]

  |0 1 0|    |0 0 0|    |0 0 1|    |0 0 0|   |0 1 0|   |1 0 1|

  [0 0 1]    [1 0 0]    [0 1 0]    [0 1 0]   [0 0 0]   [0 0 0]

MAPLE

A232568:=n->(n^6-15*n^4+28*n^3+29*n^2-76*n-15-((n+1) mod 2)*(8*n^3-21*n^2+40*n-63))/48; seq(A232568(n), n=2..50); # Wesley Ivan Hurt, Dec 06 2013

MATHEMATICA

Table[(n^6-15n^4+28n^3+29n^2-76n-15-Mod[n+1, 2](8n^3-21n^2+40n-63))/48, {n, 2, 50}] (* Wesley Ivan Hurt, Dec 06 2013 *)

CROSSREFS

Cf. A232567, A239576, A232569, A172226.

Sequence in context: A210424 A292029 A227124 * A288637 A059021 A229580

Adjacent sequences:  A232565 A232566 A232567 * A232569 A232570 A232571

KEYWORD

nonn,easy

AUTHOR

Heinrich Ludwig, Nov 28 2013

STATUS

approved

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Last modified January 18 04:47 EST 2019. Contains 319269 sequences. (Running on oeis4.)