OFFSET
0,2
COMMENTS
a(n) is the number of ordered pairs (a,b) of length n binary sequences such that a and b are equivalent by rotational symmetry. - Geoffrey Critzer, Dec 31 2011
a(n) is the weighted sum of binary strings of length n by their number of distinct images by rotation. There is a natural correspondence between the first 2^(n-1) sequences (starting with a 0) and the 2^(n-1) starting with a 1 by inversion. There is also an internal correspondance by order inversion. - Olivier Gérard, Jan 01 2011
The number of k-circulant n X n (0,1) matrices, which means the number of n X n binary matrices where rows from the 2nd row on are obtained from the preceding row by a cyclic shift by k columns for some 0 <= k < n. - R. J. Mathar, Mar 11 2017
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1000
Chuan Guo, J. Shallit, A. M. Shur, On the Combinatorics of Palindromes and Antipalindromes, arXiv preprint arXiv:1503.09112 [cs.FL], 2015.
V. V. Strok, Circulant matrices and the spectra of de Bruijn graphs, Ukr. Math. J. 44 (11) (1992) 1446-1454.
FORMULA
a(n) = Sum_{d|n} A023900(n/d)*d*2^d. - Andrew Howroyd, Sep 15 2019
EXAMPLE
a(2)= 6 because there are 6 such ordered pairs of length 2 binary sequences: (00,00),(11,11),(01,01),(10,10),(01,10),(10,01).
a(3)= 20 because the classes of 3-bit strings are 1*(000), 3*(001,010,100), 3*(011,110,101), 1*(111) = 1 + 9 + 9 + 1.
MATHEMATICA
f[n_] := 2*Plus @@ Table[ Length[ Union[ NestList[ RotateLeft, IntegerDigits[b, 2, n], n - 1]]], {b, 0, 2^(n - 1) - 1}]; f[0] = 1; Array[f, 21, 0] (* Olivier Gérard, Jan 01 2012 *)
PROG
(PARI) c(n)={sumdiv(n, d, moebius(d)*d)} \\ A023900
a(n)={if(n<1, n==0, sumdiv(n, d, c(n/d)*d*2^d))} \\ Andrew Howroyd, Sep 15 2019
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved