cp's OEIS Frontend

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

%I A063010 #43 Apr 10 2025 01:39:57
%S A063010 0,1,100,101,10000,10001,10100,10101,1000000,1000001,1000100,1000101,
%T A063010 1010000,1010001,1010100,1010101,100000000,100000001,100000100,
%U A063010 100000101,100010000,100010001,100010100,100010101,101000000,101000001
%N A063010 Carryless binary square of n; also Moser-de Bruijn sequence written in binary.
%C A063010 Numbers that are sums of distinct powers of 100. - _David Wasserman_, Feb 26 2008
%H A063010 Michael De Vlieger, <a href="/A063010/b063010.txt">Table of n, a(n) for n = 0..8191</a>
%H A063010 David Applegate, Marc LeBrun and N. J. A. Sloane, <a href="http://neilsloane.com/doc/carry1.pdf">Carryless Arithmetic (I): The Mod 10 Version</a>
%H A063010 Hsien-Kuei Hwang, Svante Janson, and Tsung-Hsi Tsai, <a href="https://arxiv.org/abs/2210.10968">Identities and periodic oscillations of divide-and-conquer recurrences splitting at half</a>, arXiv:2210.10968 [cs.DS], 2022, p. 45.
%F A063010 a(n) = A062033(n)/10, i.e., with final zero removed.
%F A063010 a(n) = Sum_{k>=0} A030308(n,k)*A098608(k). - _Philippe Deléham_, Oct 15 2011
%F A063010 G.f.: (1/(1 - x))*Sum_{k>=0} 100^k*x^(2^k)/(1 + x^(2^k)). - _Ilya Gutkovskiy_, Jun 04 2017
%e A063010 a(11)=1000101, since 11 in binary is 1011 and binary carryless sum of 1011000, 0, 10110 and 1011 is 1000101.
%t A063010 With[{k = 100}, Map[FromDigits[#, k] &, Tuples[{0, 1}, 5]]] (* _Michael De Vlieger_, Oct 29 2022 *)
%o A063010 (PARI) a(n) = fromdigits(binary(n),100); \\ _Ruud H.G. van Tol_, Dec 05 2022
%o A063010 (Python)
%o A063010 def A063010(n): return int(bin(int(bin(n)[2:],4))[2:]) # _Chai Wah Wu_, Apr 09 2025
%Y A063010 Cf. Moser-de Bruijn sequence A000695, carryless decimal squares A059729, pre-carry binary squares A063009.
%K A063010 base,easy,nonn
%O A063010 0,3
%A A063010 _Henry Bottomley_, Jul 03 2001
%E A063010 More terms from _David Wasserman_, Feb 26 2008