A274163 Number of real integers in n-th generation of tree T(4i) defined in Comments.

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

Offset: 0

Author

Clark Kimberling, Jun 12 2016

Keywords

• nonn
• more

Comments

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).

From Charlie Neder, Jul 11 2018: (Start)

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)

a(n) = A000045(n) only for 0 < n < 21. - Robert G. Wilson v, Jul 23 2018

Examples

			If r = 4i, then g(3) = {3,2r,r+1, r^2}, in which the number of real integers is a(3) = 2.
		

Crossrefs

See A274142 for a guide to related sequences.

Programs

Mathematica

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}]

Extensions

a(21)-a(25) from Robert G. Wilson v, Jul 23 2018