A187767 Number of bicolored cyclic patterns n X n.

0, 2, 3, 10, 15, 35, 63, 138, 255, 527, 1023, 2083, 4095, 8255, 16383, 32906, 65535, 131327, 262143, 524815, 1048575, 2098175, 4194303, 8390691, 16777215, 33558527, 67108863, 134225983, 268435455, 536887295, 1073741823, 2147516554, 4294967295, 8590000127, 17179869183

Offset: 1

Author

Giovanni Resta, Jan 04 2013

Keywords

• nonn

Comments

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

.

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

. 1 1 0 0 1 0 0 1 0 0 1 1 0 1 1 0

. 1 0 0 1 0 1 1 0 1 0 0 1 0 1 1 0

. 0 0 1 1 1 0 0 1 1 1 0 0 0 1 1 0

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.

Examples

			a(4)=10 is represented below. See Links for more examples.
. 1000 0100 0010 0001 0101 1010 1001 0110 1100 0011
. 0100 0001 0100 0001 0101 0101 1100 1100 0011 0011
. 0010 0100 1000 0001 0101 1010 0110 1001 1100 0011
. 0001 0001 0001 0001 0101 0101 0011 0011 0011 0011
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..200
• Giovanni Resta, Picture explaining sequence definition.
• Giovanni Resta, Pictures for a(2)-a(7).
• Giovanni Resta, Pictures for a(8) and a(9).

Crossrefs

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

Cf. A000048, A056303, A127804.

Programs

Mathematica

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]];
(*count*) a[n_] := Length@cyPatt@n; Print["Seq = ",a/@Range[12]];
(*show*) showP[p_] := GraphicsGrid@Partition[ArrayPlot/@p,8,8,1,Null];
showP[cyPatt[6]]

PARI

b(n)=sumdiv(n,d,(d%2)*(moebius(d)*2^(n/d)))/(2*n);
a(n)=sumdiv(n,d,d*b(d)) - 1; \\ Andrew Howroyd, Jun 02 2017

Formula

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

a(n) = -1 + Sum_{d|n} d*A000048(d). - Andrew Howroyd, Jun 02 2017

Extensions

a(22)-a(35) from Andrew Howroyd, Jun 02 2017