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A274144 Number of integers in n-th generation of tree T(1/4) defined in Comments.

%I A274144 #21 Jul 04 2016 03:52:49
%S A274144 1,1,1,1,1,2,2,2,2,4,4,4,5,8,8,9,12,16,17,21,27,32,37,47,57,67,82,102,
%T A274144 121,145,180,219,260,317,391,470,564,691,843,1012,1225,1500,1816,2188,
%U A274144 2663,3245,3918,4744,5778,7010,8473,10291,12511,15148
%N A274144 Number of integers in n-th generation of tree T(1/4) defined in Comments.
%C A274144 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 A274144 See A274142 for a guide to related sequences.
%H A274144 Kenny Lau, <a href="/A274144/b274144.txt">Table of n, a(n) for n = 0..11916</a>
%e A274144 For r = 1/4, we have g(3) = {3,2r,r+1, r^2}, in which only 3 is an integer, so that a(3) = 1.
%t A274144 z = 18; t = Join[{{0}}, Expand[NestList[DeleteDuplicates[Flatten[Map[{# + 1, x*#} &, #], 1]] &, {1}, z]]];
%t A274144 u = Table[t[[k]] /. x -> 1/4, {k, 1, z}];
%t A274144 Table[Count[Map[IntegerQ, u[[k]]], True], {k, 1, z}]
%Y A274144 Cf. A274142.
%K A274144 nonn,easy
%O A274144 0,6
%A A274144 _Clark Kimberling_, Jun 11 2016
%E A274144 More terms from _Kenny Lau_, Jul 01 2016