A063010 Carryless binary square of n; also Moser-de Bruijn sequence written in binary.

0, 1, 100, 101, 10000, 10001, 10100, 10101, 1000000, 1000001, 1000100, 1000101, 1010000, 1010001, 1010100, 1010101, 100000000, 100000001, 100000100, 100000101, 100010000, 100010001, 100010100, 100010101, 101000000, 101000001

Offset: 0

Author

Henry Bottomley, Jul 03 2001

Keywords

• base
• easy
• nonn

Comments

Numbers that are sums of distinct powers of 100. - David Wasserman, Feb 26 2008

Examples

			a(11)=1000101, since 11 in binary is 1011 and binary carryless sum of 1011000, 0, 10110 and 1011 is 1000101.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..8191
• David Applegate, Marc LeBrun and N. J. A. Sloane, Carryless Arithmetic (I): The Mod 10 Version
• Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, Identities and periodic oscillations of divide-and-conquer recurrences splitting at half, arXiv:2210.10968 [cs.DS], 2022, p. 45.

Crossrefs

Cf. Moser-de Bruijn sequence A000695, carryless decimal squares A059729, pre-carry binary squares A063009.

Programs

Mathematica

With[{k = 100}, Map[FromDigits[#, k] &, Tuples[{0, 1}, 5]]] (* Michael De Vlieger, Oct 29 2022 *)

PARI

a(n) = fromdigits(binary(n),100); \\ Ruud H.G. van Tol, Dec 05 2022

Python

def A063010(n): return int(bin(int(bin(n)[2:],4))[2:]) # Chai Wah Wu, Apr 09 2025

Formula

a(n) = A062033(n)/10, i.e., with final zero removed.

a(n) = Sum_{k>=0} A030308(n,k)*A098608(k). - Philippe Deléham, Oct 15 2011

G.f.: (1/(1 - x))*Sum_{k>=0} 100^k*x^(2^k)/(1 + x^(2^k)). - Ilya Gutkovskiy, Jun 04 2017

Extensions

More terms from David Wasserman, Feb 26 2008