A308465 Number of prefix normal palindromes of length n.

2, 2, 3, 3, 5, 4, 8, 7, 12, 11, 21, 18, 36, 31, 57, 55, 104, 91, 182, 166, 308, 292, 562, 512, 1009, 928, 1755, 1697, 3247, 2972, 5906, 5555, 10506, 10099, 19542, 18280, 36002, 33895, 64958, 63045, 121887, 114032, 226065, 215377, 412749, 399334, 778196, 735941

Offset: 1

Author

Michel Marcus, May 29 2019

Keywords

• nonn

Links

• Pamela Fleischmann, On Special k-Spectra, k-Locality, and Collapsing Prefix Normal Words, Ph.D. Dissertation, Kiel University (Germany, 2021).
• Pamela Fleischmann, Mitja Kulczynski, and Dirk Nowotka, On Collapsing Prefix Normal Words, arXiv:1905.11847 [cs.FL], 2019.
• Pamela Fleischmann, Mitja Kulczynski, Dirk Nowotka, and Danny Bøgsted Poulsen, On Collapsing Prefix Normal Words, Language and Automata Theory and Applications (LATA 2020) LNCS Vol. 12038, Springer, Cham, 412-424.

Crossrefs

Cf. A016116 (numbers of binary palindromes), A194850 (number of prefix normal words)

Programs

Python

from itertools import product
def is_prefix_normal(w):
  for k in range(1, len(w)+1):
    weight0 = w[:k].count("1")
    for j in range(1, len(w)-k+1):
      weightj = w[j:j+k].count("1")
      if weightj > weight0: return False
  return True
def bin_pals(digits):
  midrange = [[""], ["0", "1"]]
  for p in product("01", repeat=digits//2):
    left = "".join(p)
    for middle in midrange[digits%2]:
      yield left+middle+left[::-1]
def a(n):
  return sum(is_prefix_normal(w) for w in bin_pals(n))
print([a(n) for n in range(1, 31)]) # Michael S. Branicky, Dec 19 2020

Extensions

a(31)-a(48) from Michael S. Branicky, Dec 19 2020