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A054247
Number of n X n binary matrices under action of dihedral group of the square D_4.
38
1, 2, 6, 102, 8548, 4211744, 8590557312, 70368882591744, 2305843028004192256, 302231454921524358152192, 158456325028538104598816096256, 332306998946229005407670289177772032, 2787593149816327892769293535238052808491008
OFFSET
0,2
COMMENTS
Arises in the enumeration of "water patterns" in magic squares. [Knecht]
LINKS
Peter Kagey and William Keehn, Counting tilings of the n X m grid, cylinder, and torus, arXiv:2311.13072 [math.CO], 2023.
Craig Knecht, 102 patterns
Craig Knecht, Knecht Magic Squares Site, see sections 1 and 12.
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
(Python)
def a(n):
return 2**(n**2-3)+2**((n**2-8)/4)+2**((n**2-6)/2)+2**((n**2-4)/2)+2**((n**2+n-4)/2) if n % 2 == 0 else 2**(n**2-3)+2**((n**2-5)/4)+2**((n**2-5)/2)+2**((n**2+n-2)//2) # Peter E. Francis, Apr 12 2020
CROSSREFS
Column k=2 of A343097.
Sequence in context: A357084 A123257 A230927 * A278888 A099790 A294906
KEYWORD
easy,nonn,nice
AUTHOR
Vladeta Jovovic, May 04 2000
EXTENSIONS
More terms from Harvey P. Dale, Apr 14 2012
STATUS
approved