This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A320642 #27 Feb 16 2025 08:33:56 %S A320642 2,1,3,2,4,3,2,1,3,2,4,3,5,4,3,2,4,3,5,4,6,5,4,3,5,4,3,2,4,3,2,1,3,2, %T A320642 4,3,5,4,3,2,4,3,5,4,6,5,4,3,5,4,6,5,7,6,5,4,6,5,4,3,5,4,3,2,4,3,5,4, %U A320642 6,5,4,3,5,4,6,5,7,6,5,4,6,5,7,6,8,7,6 %N A320642 Number of 1's in the base-(-2) expansion of -n. %C A320642 Number of 1's in A212529(n). %C A320642 Define f(n) as: f(0) = 0, f(-2*n) = f(n), f(-2*n+1) = f(n) + 1, then a(n) = f(-n), n >= 1. See A027615 for the other half of f. %C A320642 For k > 1, the earliest occurrence of k is n = A086893(k-1). %H A320642 Jianing Song, <a href="/A320642/b320642.txt">Table of n, a(n) for n = 1..10000</a> %H A320642 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Negabinary.html">Negabinary</a>. %H A320642 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Negadecimal.html">Negadecimal</a>. %H A320642 Wikipedia, <a href="http://en.wikipedia.org/wiki/Negative_base">Negative base</a>. %F A320642 a(n) == -n (mod 3). %F A320642 a(n) = A000120(A005352(n)). - _Michel Marcus_, Oct 23 2018 %e A320642 A212529(11) = 110101 which has four 1's, so a(11) = 4. %e A320642 A212529(25) = 111011 which has five 1's, so a(25) = 5. %e A320642 A212529(51) = 11011101 which has six 1's, so a(51) = 6. %t A320642 b[n_] := b[n] = b[Quotient[n - 1, -2]] + Mod[n, 2]; b[0] = 0; a[n_] := b[-n]; Array[a, 100] (* _Amiram Eldar_, Jul 23 2023 *) %o A320642 (PARI) b(n) = if(n==0, 0, b(n\(-2))+n%2) %o A320642 a(n) = b(-n) %Y A320642 Cf. A000120, A005352, A027615, A086893, A212529. %K A320642 nonn,base %O A320642 1,1 %A A320642 _Jianing Song_, Oct 18 2018