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A247523 Numbers k such that d(r,k) = d(s,k), where d(x,k) = k-th binary digit of x, r = {golden ratio}, s = {(golden ratio)/2}, and { } = fractional part.

%I A247523 #16 May 05 2018 08:14:46
%S A247523 1,3,5,6,7,9,10,12,15,16,19,20,21,23,25,28,29,31,35,36,37,38,39,40,44,
%T A247523 49,51,52,53,54,56,57,58,59,65,66,67,68,70,72,73,75,77,78,80,82,84,85,
%U A247523 86,87,88,89,91,93,94,95,96,97,101,102,104,106,107,110
%N A247523 Numbers k such that d(r,k) = d(s,k), where d(x,k) = k-th binary digit of x, r = {golden ratio}, s = {(golden ratio)/2}, and { } = fractional part.
%C A247523 Every positive integer lies in exactly one of the sequences A247423 and A247524.
%H A247523 Clark Kimberling, <a href="/A247523/b247523.txt">Table of n, a(n) for n = 1..2000</a>
%e A247523 r has binary digits 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 1, ...
%e A247523 s has binary digits 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, ...
%e A247523 so that a(1) = 1 and a(2) = 3.
%t A247523 z = 400; r1 = GoldenRatio; r = FractionalPart[r1]; s = FractionalPart[r1/2];
%t A247523 u = Flatten[{ConstantArray[0, -#[[2]]], #[[1]]}] &[RealDigits[r, 2, z]];
%t A247523 v = Flatten[{ConstantArray[0, -#[[2]]], #[[1]]}] &[RealDigits[s, 2, z]];
%t A247523 t = Table[If[u[[n]] == v[[n]], 1, 0], {n, 1, z}];
%t A247523 Flatten[Position[t, 1]]  (* A247523 *)
%t A247523 Flatten[Position[t, 0]]  (* A247524 *)
%Y A247523 Cf. A247524, A247519.
%K A247523 nonn,easy,base
%O A247523 1,2
%A A247523 _Clark Kimberling_, Sep 19 2014