login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

A376697
Number of binary words of length 2^n-1 with at least n "0" between any two "1" digits.
4
1, 2, 4, 14, 106, 3970, 2951330, 601479320126, 4878266198984685082072, 20251346657999168900614712784617499550822, 2947350921470608599960387502833128388134614870362931531590353774089056633192
OFFSET
0,2
LINKS
FORMULA
a(n) = A141539(2^n-1,n).
a(n) = A376091(2^n-1).
a(n) = A376033(2^n-1,2^n-1).
a(n) = 1 + Sum_{i=0..floor((2^n-2)/(n+1))} binomial(2^n-(n*i)-1,i+1). - John Tyler Rascoe, Oct 04 2024
EXAMPLE
a(0) = 1: the empty word.
a(1) = 2: 0, 1.
a(2) = 4: 000, 100, 010, 001.
a(3) = 14: 0000000, 1000000, 0100000, 0010000, 0001000, 0000100, 1000100, 0000010, 1000010, 0100010, 0000001, 1000001, 0100001, 0010001.
PROG
(Python)
from math import comb
def A376697(n): return 1 + sum(comb(2**n-(n*i)-1, i+1) for i in range(0, (2**n-2)//(n+1)+1)) # John Tyler Rascoe, Oct 04 2024
CROSSREFS
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Oct 02 2024
STATUS
approved