A319953 List of binary words of lengths 0, 1, 2, etc., including empty word, each prefixed by a 2.

2, 20, 21, 200, 201, 210, 211, 2000, 2001, 2010, 2011, 2100, 2101, 2110, 2111, 20000, 20001, 20010, 20011, 20100, 20101, 20110, 20111, 21000, 21001, 21010, 21011, 21100, 21101, 21110, 21111, 200000, 200001, 200010, 200011, 200100, 200101, 200110, 200111

Offset: 0

Author

N. J. A. Sloane, Oct 04 2018

Keywords

• nonn
• base

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..25000
• Carl Pomerance, John Michael Robson, and Jeffrey Shallit, Automaticity II: Descriptional complexity in the unary case, Theoretical computer science 180.1-2 (1997): 181-201.

Crossrefs

Cf. A000523, A007088, A214339, A319954.

Programs

PARI

a(n) = my (b=binary(n+1)); b[1]++; fromdigits(b) \\ Rémy Sigrist, Oct 04 2018

Python

def a(n): return int(bin(n)[2:].replace('1', '2', 1))
print([a(n) for n in range(1, 40)]) # Michael S. Branicky, Feb 20 2021

Formula

a(n) = A007088(n+1) + 10^A000523(n+1). - Rémy Sigrist, Oct 04 2018

Extensions

Data corrected and extended by Rémy Sigrist, Oct 04 2018