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A380203 With given points 0,1 on the x-axis, a(n) is the number of ways to construct n with m circles where 2^(m-1)

%I A380203 #17 Feb 07 2025 14:18:39
%S A380203 1,1,1,1,1,2,1,1,2,4,2,4,1,2,1,1,5,9,6,10,4,8,4,8,1,4,2,4,1,2,1,1,15,
%T A380203 28,15,31,13,25,14,28,10,19,11,22,8,15,9,17,2,8,4,12,2,8,4,8,1,4,2,4,
%U A380203 1,2,1,1,50,94,56,99,45,91,51,97,39,74,41,92,31,74,40,85,26,61
%N A380203 With given points 0,1 on the x-axis, a(n) is the number of ways to construct n with m circles where 2^(m-1)<n<=2^m.
%C A380203 Description of the construction: The first circle with center 1 and radius 1 intersects the x-axis at 2. The center of the next circle is 2. Each circle, except for the last one, intersects the axis at the center of the next one. The last circle intersects the axis at x.
%C A380203 The sequence of indices with a(n)=1 is A379972. Further comments on the construction can be found there. The algorithm generates a tree. In this section, a(6)=2 and a(n)=1 otherwise.
%C A380203         5
%C A380203        /
%C A380203       3
%C A380203      / \
%C A380203   1-2   6
%C A380203      \ /
%C A380203       4-7
%C A380203        \
%C A380203         8
%e A380203  n a(n)  intersection points on the x-axis
%e A380203  5  1    2, 3, 5
%e A380203  6  2    2, 3, 6 or 2, 4, 6
%e A380203 10  4    2, 3, 5, 10 or 2, 3, 6, 10 or 2, 4, 6, 10 or 2, 4, 7, 10
%o A380203 (Maxima)
%o A380203 block(m:7, aa:makelist(i-1,i,1,m+2), freq:makelist(0,i,1,2^m),
%o A380203 recursion(t):=  (freq[aa[t]] : freq[aa[t]] + 1,
%o A380203  if t< m+2 then
%o A380203    for k from t-1 thru 1 step -1 do
%o A380203      (p:2*aa[t] - aa[k] ,
%o A380203      if p> 2^(t-2) then (aa[t+1]:p, recursion(t+1) ) ) ) ,
%o A380203  recursion(2), freq );
%Y A380203 Cf. A379972.
%K A380203 nonn
%O A380203 1,6
%A A380203 _Gerhard Kirchner_, Jan 16 2025