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A054247 Number of n X n binary matrices under action of dihedral group of the square D_4. 27
1, 2, 6, 102, 8548, 4211744, 8590557312, 70368882591744, 2305843028004192256, 302231454921524358152192, 158456325028538104598816096256, 332306998946229005407670289177772032, 2787593149816327892769293535238052808491008 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Arises in the enumeration of "water patterns" in magic squares. [Knecht]

LINKS

Table of n, a(n) for n=0..12.

Craig Knecht, 102 patterns

Craig Knecht, Knecht Magic Squares Site, see sections 1 and 12.

Index entries for sequences related to groups

FORMULA

a(n) = (1/8)*(2^(n^2)+2*2^(n^2/4)+3*2^(n^2/2)+2*2^((n^2+n)/2)) if n is even and a(n) = (1/8)*(2^(n^2)+2*2^((n^2+3)/4)+2^((n^2+1)/2)+4*2^((n^2+n)/2)) if n is odd.

EXAMPLE

There are 6 nonisomorphic 2 X 2 matrices under action of D_4:

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

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

MATHEMATICA

f[n_]:=With[{n2=n^2}, If[EvenQ[n], (2^n2+2(2^(n2/4))+3(2^(n2/2))+ 2(2^((n2+n)/2)))/8, (2^n2+2(2^((n2+3)/4))+2^((n2+1)/2)+ 4(2^((n2+n)/2)))/8]]; Array[f, 15, 0] (* Harvey P. Dale, Apr 14 2012 *)

PROG

(PARI) a(n)=(2^n^2+2^((n^2+7)\4)+if(n%2, 2^((n^2+1)/2)+2^((n^2+n+4)/2), 3*2^(n^2/2)+2^((n^2+n+2)/2)))/8 \\ Charles R Greathouse IV, May 27 2014

CROSSREFS

Cf. A002724, A054407.

Sequence in context: A100704 A123257 A230927 * A278888 A099790 A284262

Adjacent sequences:  A054244 A054245 A054246 * A054248 A054249 A054250

KEYWORD

easy,nonn,nice

AUTHOR

Vladeta Jovovic, May 04 2000

EXTENSIONS

More terms from Harvey P. Dale, Apr 14 2012

STATUS

approved

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Last modified May 23 18:23 EDT 2017. Contains 286926 sequences.