

A045655


Number of 2nbead balanced binary strings, rotationally equivalent to reversed complement.


11



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
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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^(n1) sequences (starting with a 0) and the 2^(n1) starting with a 1 by inversion. There is also an internal correspondance by order inversion.  Olivier Gérard, Jan 01 2011
The number of kcirculant 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



FORMULA



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 3bit 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

Cf. A000031 counts the string classes.


KEYWORD

nonn


AUTHOR



STATUS

approved



