A054247
Number of n X n binary matrices under action of dihedral group of the square D_4.
Original entry on oeis.org
1, 2, 6, 102, 8548, 4211744, 8590557312, 70368882591744, 2305843028004192256, 302231454921524358152192, 158456325028538104598816096256, 332306998946229005407670289177772032, 2787593149816327892769293535238052808491008
Offset: 0
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].
- Peter E. Francis, Table of n, a(n) for n = 0..57
- 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.
- Index entries for sequences related to groups
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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 *)
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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
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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
A140650
Number of different ways of coloring an n X n grid of squares using two colors so that the resulting grid has just one line of symmetry.
Original entry on oeis.org
0, 3, 48, 600, 32256, 1177344, 268369920, 36506664960, 35184338534400, 18577347909255168, 73786976226118729728, 153476910691030086451200, 2475880078570197599844827136, 20440865928680162788862343512064, 1329227995784915854457062986570792960
Offset: 1
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s=[0]; for(n=1, 10, s=concat(s, [2^(2*n^2-1)*(2^n+1)-2^(n^2-1)*(2^n+1), 2^(2*n^2+3*n+1)-2^(n^2+2*n+1)])); s \\ Colin Barker, Mar 28 2014
Original entry on oeis.org
1, 3, 9, 111, 8659, 4220403, 8594777715, 70377477369459, 2305913405481561715, 302233760834929839713907, 158456627262298939528655810163, 332307157402856267706609817833582195
Offset: 0
a(4) = 1 + 2 + 6 + 102 + 8548 = 8659 = 7 * 1237.
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A054247(n)={ local(a) ; if(n%2==0, a=2^(n^2)+2*2^(n^2/4)+3*2^(n^2/2)+2*2^((n^2+n)/2), a=2^(n^2)+2*2^((n^2+3)/4)+2^((n^2+1)/2)+4*2^((n^2+n)/2); ) ; return(a/8) ; }
A177793(n)={ return(sum(i=0,n,A054247(i))) ; }
{ for(n=0,20, print1(A177793(n),",") ; ) ; } (End)
A140795
We consider how many ways there are of coloring a square grid, n X n, using just two colors, black & white say. If the resulting grid has rotational symmetry of order two only, then the number of different grids is given by this sequence. None of these counted are the images of any of the others under a reflection or a rotation of 90 degrees. If one wishes to count these as different, then each of these numbers can be multiplied by 4.
Original entry on oeis.org
0, 0, 2, 44, 1792, 64288, 8354304, 1073447424, 549738528768, 281474691514368, 576460717407862784, 1180591619583540985856, 9671406556633359531900928, 79228162514246041720191975424, 2596148429267404554864448650608640, 85070591730234614676028659138035712000
Offset: 1
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s=[0]; for(m=1, 15, s=concat(s, [(2^(2*m^2)-2^m^2*(2^m+2)+2^((m^2+m+2)/2))/4, 2^(2*m^2+2*m-1)-2^(m^2-1)*(2^(2*m+1)+2^m)+2^(m*(m+3)/2)])); s \\ Colin Barker, Mar 28 2014
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