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 A330882 Number of length-n binary strings having the longest possible LB factorization. 2
 1, 2, 2, 4, 2, 4, 4, 32, 14, 28, 8, 16, 4, 8, 8, 176, 48, 96, 20, 40, 8, 16, 16, 640, 140, 280, 48, 96, 16, 32, 32, 1992, 376, 752, 112, 224, 32, 64, 64, 5696, 960, 1920 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS A border of a string w is a nonempty proper prefix of w that is also a suffix.  The LB ("longest border") factorization of a string w is as follows:  if w has no border, then the factorization is just (w).  Otherwise, write w = (x)(w')(x) where x is the longest border of length <= |w|/2, and continue with w'.   The length of the factorization is the number of factors.  For example, 0101101010 = (010)(1)(10)(1)(010), and so has length 5. LINKS PROG (Python) from numba import njit @njit()  # comment out for digits > 64 def LBfactors(w, digits):   if digits <= 1: return digits   if not (1 << (digits-1)) & w:  # if the 1st bit is not 1,     w ^= ((1 << digits) - 1)     # then invert the string   for i in range(digits//2, 0, -1):     mask = (1 << i) - 1     if (w >> (digits-i)) == (w & mask):       digitsprime = digits - 2*i       if digitsprime == 0:         return 2       else:         middle_mask = ((1 << digitsprime) - 1)         wprime = middle_mask & (w >> i)         return 2 + LBfactors(wprime, digitsprime)   return 1 @njit()  # comment out for n > 64 def a(n):   if n <= 1: return 2**n   maximum, maximum_count = -1, 0   for i in range(2**(n-1)):  # only search 1st bit == 1 by symmetry     LBfacsw = LBfactors((1<<(n-1))|i, n)     if LBfacsw == maximum:       maximum_count += 1     elif LBfacsw > maximum:       maximum = LBfacsw       maximum_count = 1   return 2*maximum_count     # symmetry print([a(n) for n in range(25)]) # Michael S. Branicky, Dec 31 2020 CROSSREFS Cf. A330881, A330884. Sequence in context: A079314 A323381 A060609 * A205138 A233763 A109526 Adjacent sequences:  A330879 A330880 A330881 * A330883 A330884 A330885 KEYWORD nonn,more AUTHOR Jeffrey Shallit, Apr 30 2020 EXTENSIONS a(28)-a(41) from Michael S. Branicky, Dec 31 2020 STATUS approved

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Last modified August 4 21:32 EDT 2021. Contains 346455 sequences. (Running on oeis4.)