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

0,3

Comments

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

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.

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

Example

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

Mathematica

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

Prog

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

Crossrefs

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

Sequence in context: A323823 A085251 A178530 * A215022 A094027 A281149

Adjacent sequences: A063007 A063008 A063009 * A063011 A063012 A063013

Keyword

base,easy,nonn

Author

Henry Bottomley, Jul 03 2001

Extensions

More terms from David Wasserman, Feb 26 2008

Status

approved