OFFSET
1,2
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.
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).
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
EXAMPLE
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
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]]
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Giovanni Resta, Jan 04 2013
EXTENSIONS
a(22)-a(35) from Andrew Howroyd, Jun 02 2017
STATUS
approved