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%I #20 Aug 30 2018 15:50:50
%S 1,1,1,2,3,5,8,13,21,34,55,89,144,233,377,610,987,1597,2584,4181,6766,
%T 10948,17716,28667,46388,75063
%N Number of real integers in n-th generation of tree T(4i) defined in Comments.
%C Let T* be the infinite tree with root 0 generated by these rules: if p is in T*, then p+1 is in T* and x*p is in T*. Let g(n) be the set of nodes in the n-th generation, so that g(0) = {0}, g(1) = {1}, g(2) = {2,x}, g(3) = {3,2x,x+1,x^2}, etc. Let T(r) be the tree obtained by substituting r for x.
%C For each integer k > 0, let s(k,n) be the number of integers in the n-th generation of T(k*i). Conjecture: there is a limiting sequence S(n) as k increases, and S(n) = F(n) for n >= 1, where F = A000045 (Fibonacci numbers).
%C From _Charlie Neder_, Jul 11 2018: (Start)
%C Assume for the moment that a complex number cannot be transformed back into an integer. If this is the case, then the real integers in g(n) are the real integers in g(n-1) plus 1 and the imaginary integers in g(n-1) times k*i, which are themselves k*i times the real integers in g(n-2), and so S(n) = S(n-1) + S(n-2) and S(n) = F(n).
%C However, the above assumption is false, but the earliest time such a transformation can take place is at g(k^2+5), following this path: 0 -> 1 -> k*i -> 1+k*i -> -k^2+k*i -> -(k^2-1)+k*i -> ... -> k*i -> -k^2.
%C Therefore s(k,n) matches the Fibonacci sequence for n < k^2+5 and S(n) = F(n). (End)
%C a(n) = A000045(n) only for 0 < n < 21. - _Robert G. Wilson v_, Jul 23 2018
%e If r = 4i, then g(3) = {3,2r,r+1, r^2}, in which the number of real integers is a(3) = 2.
%t z = 18; t = Join[{{0}}, Expand[NestList[DeleteDuplicates[Flatten[Map[{# + 1, x*#} &, #], 1]] &, {1}, z]]];
%t u = Table[t[[k]] /. x -> 4 I, {k, 1, z}]; Table[Count[Map[IntegerQ, u[[k]]], True], {k, 1, z}]
%Y See A274142 for a guide to related sequences.
%K nonn,more
%O 0,4
%A _Clark Kimberling_, Jun 12 2016
%E a(21)-a(25) from _Robert G. Wilson v_, Jul 23 2018
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