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

1,1

Links

Table of n, a(n) for n=1..48.

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.

Prog

(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

Crossrefs

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

Sequence in context: A114328 A097366 A139807 * A276119 A167755 A259788

Adjacent sequences: A308462 A308463 A308464 * A308466 A308467 A308468

Keyword

nonn

Author

Michel Marcus, May 29 2019

Extensions

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

Status

approved