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A274163
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Number of real integers in n-th generation of tree T(4i) defined in Comments.
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1
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1, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6766, 10948, 17716, 28667, 46388, 75063
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OFFSET
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0,4
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COMMENTS
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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.
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).
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).
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.
Therefore s(k,n) matches the Fibonacci sequence for n < k^2+5 and S(n) = F(n). (End)
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LINKS
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EXAMPLE
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If r = 4i, then g(3) = {3,2r,r+1, r^2}, in which the number of real integers is a(3) = 2.
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MATHEMATICA
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z = 18; t = Join[{{0}}, Expand[NestList[DeleteDuplicates[Flatten[Map[{# + 1, x*#} &, #], 1]] &, {1}, z]]];
u = Table[t[[k]] /. x -> 4 I, {k, 1, z}]; Table[Count[Map[IntegerQ, u[[k]]], True], {k, 1, z}]
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CROSSREFS
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See A274142 for a guide to related sequences.
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KEYWORD
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nonn,more
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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