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 A045655 Number of 2n-bead 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 (list; graph; refs; listen; history; text; internal format)
 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 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 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 = 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. Cf. A000984, A023900, A045664, A045653, A045654, A045656. Sequence in context: A060344 A363600 A347582 * A303307 A321192 A327414 Adjacent sequences: A045652 A045653 A045654 * A045656 A045657 A045658 KEYWORD nonn AUTHOR David W. Wilson STATUS approved

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Last modified September 29 03:45 EDT 2023. Contains 365749 sequences. (Running on oeis4.)