A274158 Number of integers in n-th generation of tree T(2^(-1/3)) defined in Comments.

1, 1, 1, 1, 1, 2, 2, 3, 3, 5, 6, 8, 9, 13, 17, 22, 27, 36, 47, 61, 77, 101, 132, 171, 219, 285, 370, 480, 619, 803, 1042, 1351, 1747, 2264, 2936, 3805, 4927, 6385, 8276, 10725, 13894, 18004, 23333, 30238, 39179, 50770, 65794, 85261, 110483, 143171, 185534, 240432, 311566, 403749, 523216, 678031

Offset: 0

Author

Clark Kimberling, Jun 12 2016

Keywords

• nonn
• easy

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.

See A274142 for a guide to related sequences.

Examples

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

Links

• Kenny Lau, Table of n, a(n) for n = 0..8877

Crossrefs

Cf. A274142.

Programs

Mathematica

z = 18; t = Join[{{0}}, Expand[NestList[DeleteDuplicates[Flatten[Map[{# + 1, x*#} &, #], 1]] &, {1}, z]]];
u = Table[t[[k]] /. x -> 2^(-1/3), {k, 1, z}]; Table[Count[Map[IntegerQ, u[[k]]], True], {k, 1, z}]

Extensions

More terms from Kenny Lau, Jul 04 2016