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A274159 Number of integers in n-th generation of tree T(3^(-1/3)) defined in Comments.

%I A274159 #17 Oct 02 2017 03:59:28
%S A274159 1,1,1,1,1,1,2,2,2,3,3,4,5,6,7,9,11,12,16,18,23,28,33,41,49,61,72,89,
%T A274159 107,130,159,191,234,283,345,418,507,617,747,910,1103,1340,1629,1976,
%U A274159 2402,2914,3542,4300,5223,6344,7701,9359,11361,13801,16761,20353,24725,30021,36468,44285,53788,65328
%N A274159 Number of integers in n-th generation of tree T(3^(-1/3)) defined in Comments.
%C A274159 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 A274159 See A274142 for a guide to related sequences.
%H A274159 Kenny Lau, <a href="/A274159/b274159.txt">Table of n, a(n) for n = 0..11841</a>
%e A274159 If r = 3^(-1/3), then g(3) = {3,2r,r+1, r^2}, in which the number of integers is a(3) = 1.
%p A274159 A274159 := proc(r)
%p A274159     local gs, n, gs2, el, a ;
%p A274159     gs := [1] ;
%p A274159     for n from 2 do
%p A274159         gs2 := [] ;
%p A274159         for el in gs do
%p A274159             gs2 := [op(gs2), el+1, r*el] ;
%p A274159         end do:
%p A274159         gs := gs2 ;
%p A274159         a := 0 ;
%p A274159         for el in gs do
%p A274159             if type(el, 'integer') then
%p A274159                  a := a+1 :
%p A274159             end if;
%p A274159         end do:
%p A274159         print(n, a) ;
%p A274159     end do:
%p A274159 end proc:
%p A274159 A274159(1/root[3](3)) ; # _R. J. Mathar_, Jun 20 2016
%t A274159 z = 18; t = Join[{{0}}, Expand[NestList[DeleteDuplicates[Flatten[Map[{# + 1, x*#} &, #], 1]] &, {1}, z]]];
%t A274159 u = Table[t[[k]] /. x -> 3^(-1/3), {k, 1, z}]; Table[Count[Map[IntegerQ, u[[k]]], True], {k, 1, z}]
%Y A274159 Cf. A274142.
%K A274159 nonn
%O A274159 0,7
%A A274159 _Clark Kimberling_, Jun 12 2016
%E A274159 a(15)-a(18) from _R. J. Mathar_, Jun 20 2016
%E A274159 More terms from _Kenny Lau_, Jul 04 2016