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A156025
Number of n-bit numbers whose binary representation's substrings represent the maximal number (A112509(n)) of distinct integers.
6
2, 1, 1, 3, 2, 6, 5, 1, 4, 5, 2, 8, 10, 4, 16, 22, 12, 2, 10, 19, 17, 7, 1, 5, 9, 7, 2, 11, 24, 28, 20
OFFSET
1,1
COMMENTS
If only positive integer substrings are counted (see A156022), the first two terms become 1,2 (the n-bit numbers in question being 1, 10, 11 in binary) and all subsequent terms are unchanged.
LINKS
2008/9 British Mathematical Olympiad Round 2, Problem 4, Jan 29 2009.
EXAMPLE
The n-bit numbers in question are, in binary, for n <= 8: 0 1; 10; 110; 1100 1101 1110, 11100 11101; 111000 111001 111010 111011 111100 111101; 1110100 1111000 1111001 1111010 1111011; 11110100.
PROG
(Python)
from itertools import product
def c(w):
return len(set(w[i:j+1] for i in range(len(w)) if w[i] != "0" for j in range(i, len(w)))) + int("0" in w)
def a(n):
if n == 1: return 2
m, argm, cardm = -1, None, 0
for b in product("01", repeat=n-1):
v = c("1"+"".join(b))
if v == m: argm, cardm = int("1"+"".join(b), 2), cardm + 1
elif v > m: m, argm, cardm = v, int("1"+"".join(b), 2), 1
return cardm
print([a(n) for n in range(1, 21)]) # Michael S. Branicky, Jan 13 2023
CROSSREFS
Cf. A078822, A112509 (corresponding maximum), A112510 (least such number), A112511 (greatest such number), A122953, A156022, A156023, A156024.
Sequence in context: A205115 A334369 A093623 * A035556 A296086 A122044
KEYWORD
nonn,base,more
AUTHOR
Joseph Myers, Feb 01 2009
STATUS
approved