A274145 Number of integers in n-th generation of tree T(2/3) defined in Comments.

1, 1, 1, 1, 2, 2, 3, 4, 5, 6, 9, 11, 14, 19, 25, 32, 43, 56, 73, 97, 128, 167, 222, 292, 384, 507, 670, 882, 1165, 1539, 2030, 2680, 3541, 4675, 6173, 8155, 10772, 14227, 18798, 24834, 32808, 43350, 57279, 75681, 100006, 132152, 174627, 230766, 304963, 403012, 532600, 703874, 930227, 1229386

Offset: 0

Author

Clark Kimberling, Jun 11 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

			For r = 2/3, we have g(3) = {3,2r,r+1, r^2}, in which only 3 is an integer, so that a(3) = 1.
		

Links

• Kenny Lau, Table of n, a(n) for n = 0..8248

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 -> 2/3, {k, 1, z}];
Table[Count[Map[IntegerQ, u[[k]]], True], {k, 1, z}]

Extensions

More terms from Kenny Lau, Jul 01 2016