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A379972 Numbers x with 2^(m-1) which can unambiguously be constructed with given points 0,1 on the x-axis and m circles.

%I A379972 #12 Jan 15 2025 08:49:09
%S A379972 1,2,3,4,5,7,8,13,15,16,25,29,31,32,57,61,63,64,113,121,125,127,128,
%T A379972 241,249,253,255,256,481,497,505,509,511,512,993,1009,1017,1021,1023,
%U A379972 1024,1985,2017,2033,2041,2045,2047,2048
%N A379972 Numbers x with 2^(m-1)<x<=2^m which can unambiguously be constructed with given points 0,1 on the x-axis and m circles.
%C A379972 Each circle, except for the last one, intersects the x-axis at the center of the next one. Any x>1 is represented by intersection points: y(1)=2, y(2),..., y(m)=x, but not necessarily in a unique way. The standard representation can be found by a backward algorithm: If y(j) is even, y(j-1)= y(j)/2, otherwise y(j)=(y(j)+1)/2. This way, only circles intersecting the x-axis at 0 or 1 are used. If no other representation exists, x belongs to the sequence, see examples.
%C A379972 Further comments, proof of the formula and images, see link "Construction with circles".
%H A379972 Gerhard Kirchner, <a href="/A379972/a379972.pdf">Construction with circles</a>
%F A379972 x=2^m-2^k+1 with m>=0 and 0<=k<=(m+1)/2.
%F A379972 For x=1, only m=0 makes sense. Therefore k=m=1 is excluded.
%e A379972 Example 1: k=0, x=2^m belongs to the sequence.
%e A379972  Standard repesentation: (2,..,2^j,..,2^m)
%e A379972 Example 2: m=6, k=3, x=57 belongs to the sequence.
%e A379972  Standard repesentation: (2,4,8,15,29,57)
%e A379972 Counterexample 3: m=6, k=4, x=49 does not belong to the sequence.
%e A379972  Standard repesentation: (2,4,7,13,25,49)
%e A379972  Other repesentation:    (2,4,7,14,28,49)
%e A379972 Counterexample 4: x=48 does not belong to the sequence.
%e A379972  Standard repesentation: (2,3,6,12,24,48)
%e A379972  Other repesentation:    (2,4,7,13,25,48)
%o A379972 (Maxima)
%o A379972 block(u:[],
%o A379972 for m from 0 thru 11 do
%o A379972   for k from floor((m+1)/2) thru 0 step -1 do
%o A379972     if m=0 or k<m then
%o A379972       (x : 2 ^ m - 2 ^ k + 1, u:append(u,[x])), u);
%K A379972 nonn
%O A379972 1,2
%A A379972 _Gerhard Kirchner_, Jan 08 2025