A045655 Number of 2n-bead balanced binary strings, rotationally equivalent to reversed complement.

1, 2, 6, 20, 54, 152, 348, 884, 1974, 4556, 10056, 22508, 48636, 106472, 228444, 491120, 1046454, 2228192, 4713252, 9961436, 20960904, 44038280, 92252100, 192937940, 402599676, 838860152, 1744723896, 3623869388, 7515962172

Offset: 0

Author

David W. Wilson

Keywords

• nonn

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

Examples

			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.
		

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.

Crossrefs

Cf. A000031 counts the string classes.

Cf. A000984, A023900, A045664, A045653, A045654, A045656.

Programs

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 *)

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

Formula

For n >= 1, a(n) = Sum_{d|n} A045664(d) = Sum_{d|n} d*A027375(d) = Sum_{d|n} d^2*A001037(d).

a(n) = Sum_{d|n} A023900(n/d)*d*2^d. - Andrew Howroyd, Sep 15 2019