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 A158705 #46 Feb 16 2025 08:33:10
%S A158705 1,3,4,6,9,11,12,14,16,18,21,23,24,26,29,31,33,35,36,38,41,43,44,46,
%T A158705 48,50,53,55,56,58,61,63,64,66,69,71,72,74,77,79,81,83,84,86,89,91,92,
%U A158705 94,96,98,101,103,104,106,109,111,113,115,116,118,121,123,124,126
%N A158705 Nonnegative integers with an odd number of even powers of 2 in their base-2 representation.
%C A158705 The nonnegative integers with an even number of even powers of 2 in their base-2 representation are given in A158704.
%C A158705 It appears that a result similar to Prouhet's theorem holds for the terms of A158704 and A158705, specifically: Sum_{k=0..2^n-1, k has an even number of even powers of 2} k^j = Sum_{k=0..2^n-1, k has an odd number of even powers of 2} k^j, for 0 <= j <= (n-1)/2. For a recent treatment of this theorem, see the reference.
%C A158705 Take any binary vector of length 4n+1 with n >= 0. We can activate any bits. When a bit is activated, neighboring bits change their values 0 -> 1, 1 -> 0. Our goal is to turn the original binary vector into a vector of only ones by activating the bits. If the value of the binary vector belongs to this sequence, this is possible for a maximum of 4n+1 activations. - _Mikhail Kurkov_, Jun 03 2021 [verification needed]
%H A158705 Amiram Eldar, <a href="/A158705/b158705.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Harvey P. Dale)
%H A158705 Chris Bernhardt, <a href="http://www.jstor.org/stable/27643161">Evil twins alternate with odious twins</a>, Math. Mag. 82 (2009), pp. 57-62.
%H A158705 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Prouhet-Tarry-EscottProblem.html">Prouhet-Tarry-Escott Problem</a>
%H A158705 <a href="/index/Ar#2-automatic">Index entries for 2-automatic sequences</a>.
%e A158705 The base-2 representation of 6 is 110, i.e., 6 = 2^2 + 2^1, with one even power of 2. Thus 6 is a term of the sequence.
%t A158705 Select[Range[100],OddQ[Total[Take[Reverse[IntegerDigits[#,2]],{1,-1,2}]]]&] (* _Harvey P. Dale_, Dec 23 2012 *)
%o A158705 (Magma) [ n : n in [0..150] | IsOdd(&+Intseq(n, 4))]; // _Vincenzo Librandi_, Apr 13 2011
%Y A158705 Cf. A341389 (characteristic function), A158704 (complement).
%Y A158705 Cf. A000069, A001969, A157971.
%K A158705 nonn,base
%O A158705 1,2
%A A158705 _John W. Layman_, Mar 26 2009