%I #47 Jan 31 2021 20:08:00

%S 1,2,6,20,54,152,348,884,1974,4556,10056,22508,48636,106472,228444,

%T 491120,1046454,2228192,4713252,9961436,20960904,44038280,92252100,

%U 192937940,402599676,838860152,1744723896,3623869388,7515962172

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

%C 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

%C 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

%C 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

%H Alois P. Heinz, <a href="/A045655/b045655.txt">Table of n, a(n) for n = 0..1000</a>

%H Chuan Guo, J. Shallit, A. M. Shur, <a href="http://arxiv.org/abs/1503.09112">On the Combinatorics of Palindromes and Antipalindromes</a>, arXiv preprint arXiv:1503.09112 [cs.FL], 2015.

%H V. V. Strok, <a href="http://dx.doi.org/ 10.1007/BF01071520">Circulant matrices and the spectra of de Bruijn graphs</a>, Ukr. Math. J. 44 (11) (1992) 1446-1454.

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

%F a(n) = Sum_{d|n} A023900(n/d)*d*2^d. - _Andrew Howroyd_, Sep 15 2019

%e 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).

%e 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.

%t 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 *)

%o (PARI) c(n)={sumdiv(n,d, moebius(d)*d)} \\ A023900

%o a(n)={if(n<1, n==0, sumdiv(n, d, c(n/d)*d*2^d))} \\ _Andrew Howroyd_, Sep 15 2019

%Y Cf. A000031 counts the string classes.

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

%K nonn

%O 0,2

%A _David W. Wilson_