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

%I #20 Jul 04 2016 03:53:43

%S 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,

%T 507,670,882,1165,1539,2030,2680,3541,4675,6173,8155,10772,14227,

%U 18798,24834,32808,43350,57279,75681,100006,132152,174627,230766,304963,403012,532600,703874,930227,1229386

%N Number of integers in n-th generation of tree T(2/3) 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 See A274142 for a guide to related sequences.

%H Kenny Lau, <a href="/A274145/b274145.txt">Table of n, a(n) for n = 0..8248</a>

%e 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.

%t z = 18; t = Join[{{0}}, Expand[NestList[DeleteDuplicates[Flatten[Map[{# + 1, x*#} &, #], 1]] &, {1}, z]]];

%t u = Table[t[[k]] /. x -> 2/3, {k, 1, z}];

%t Table[Count[Map[IntegerQ, u[[k]]], True], {k, 1, z}]

%Y Cf. A274142.

%K nonn,easy

%O 0,5

%A _Clark Kimberling_, Jun 11 2016

%E More terms from _Kenny Lau_, Jul 01 2016