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A187767 Number of bicolored cyclic patterns n X n. 2

%I

%S 0,2,3,10,15,35,63,138,255,527,1023,2083,4095,8255,16383,32906,65535,

%T 131327,262143,524815,1048575,2098175,4194303,8390691,16777215,

%U 33558527,67108863,134225983,268435455,536887295,1073741823,2147516554,4294967295,8590000127,17179869183

%N Number of bicolored cyclic patterns n X n.

%C A bicolored cyclic pattern is a 0-1 n x n matrix where the j-th row is equal to the first row rotated to the left by (j-1)*k places, with 1 <= k <= n a parameter. For example, with first row = 0110 we have

%C .

%C . (k=1) 0 1 1 0 (k=2) 0 1 1 0 (k=3) 0 1 1 0 (k=4) 0 1 1 0

%C . 1 1 0 0 1 0 0 1 0 0 1 1 0 1 1 0

%C . 1 0 0 1 0 1 1 0 1 0 0 1 0 1 1 0

%C . 0 0 1 1 1 0 0 1 1 1 0 0 0 1 1 0

%C The (2^n-2)*n matrices so obtained are reduced considering equivalent those obtained exchanging 0's and 1's and those which produce the same pattern, apart translation.

%H Andrew Howroyd, <a href="/A187767/b187767.txt">Table of n, a(n) for n = 1..200</a>

%H Giovanni Resta, <a href="/A187767/a187767.png">Picture explaining sequence definition.</a>

%H Giovanni Resta, <a href="/A187767/a187767_1.png">Pictures for a(2)-a(7).</a>

%H Giovanni Resta, <a href="/A187767/a187767_2.png">Pictures for a(8) and a(9).</a>

%F a(1) = 0; a(n) = 2^(n-1)-1 if n is odd, 2^(n-1)+a(n/2) if n is even (conjectured).

%F a(n) = -1 + Sum_{d|n} d*A000048(d). - _Andrew Howroyd_, Jun 02 2017

%e a(4)=10 is represented below. See Links for more examples.

%e . 1000 0100 0010 0001 0101 1010 1001 0110 1100 0011

%e . 0100 0001 0100 0001 0101 0101 1100 1100 0011 0011

%e . 0010 0100 1000 0001 0101 1010 0110 1001 1100 0011

%e . 0001 0001 0001 0001 0101 0101 0011 0011 0011 0011

%t cyPatt[n_]:=Block[{b,c},c[v_,q_:1]:=Table[RotateLeft[v,i q],{i,n}]; b=Union[(First@Union[c@#,c[1-#]])& /@ IntegerDigits[Range[2^n/2-1], 2,n]]; Union@Flatten[Table[c[e,j],{j,n},{e,b}],1]];

%t (*count*) a[n_] := Length@cyPatt@n; Print["Seq = ",a/@Range[12]];

%t (*show*) showP[p_] := GraphicsGrid@Partition[ArrayPlot/@p,8,8,1,Null];

%t showP[cyPatt[6]]

%o (PARI)

%o b(n)=sumdiv(n,d,(d%2)*(moebius(d)*2^(n/d)))/(2*n);

%o a(n)=sumdiv(n,d,d*b(d)) - 1; \\ _Andrew Howroyd_, Jun 02 2017

%Y The number of patterns made of vertical stripes only is A056295(n).

%Y Cf. A000048, A056303, A127804.

%K nonn

%O 1,2

%A _Giovanni Resta_, Jan 04 2013

%E a(22)-a(35) from _Andrew Howroyd_, Jun 02 2017

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Last modified October 19 23:44 EDT 2019. Contains 328244 sequences. (Running on oeis4.)