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%I #45 Apr 28 2024 09:50:13
%S 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 48,50,53,55,56,58,61,63,64,66,69,71,72,74,77,79,81,83,84,86,89,91,92,
%U 94,96,98,101,103,104,106,109,111,113,115,116,118,121,123,124,126
%N Nonnegative integers with an odd number of even powers of 2 in their base-2 representation.
%C The nonnegative integers with an even number of even powers of 2 in their base-2 representation are given in A158704.
%C 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 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 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 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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Prouhet-Tarry-EscottProblem.html">Prouhet-Tarry-Escott Problem</a>
%H <a href="/index/Ar#2-automatic">Index entries for 2-automatic sequences</a>.
%e 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 Select[Range[100],OddQ[Total[Take[Reverse[IntegerDigits[#,2]],{1,-1,2}]]]&] (* _Harvey P. Dale_, Dec 23 2012 *)
%o (Magma) [ n : n in [0..150] | IsOdd(&+Intseq(n, 4))]; // _Vincenzo Librandi_, Apr 13 2011
%Y Cf. A341389 (characteristic function), A158704 (complement).
%Y Cf. A000069, A001969, A157971.
%K nonn,base
%O 1,2
%A _John W. Layman_, Mar 26 2009
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