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

%I #16 Jul 04 2016 05:58:25

%S 1,1,1,1,2,2,2,4,4,5,8,9,12,16,20,26,34,44,57,74,97,125,162,212,272,

%T 356,462,597,780,1010,1311,1706,2210,2873,3732,4841,6294,8168,10608,

%U 13781,17886,23237,30172,39177,50891,66072,85813,111446,144706,187947,244059,316937,411618,534503,694153,901461

%N Number of integers in n-th generation of tree T(1/3) 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="/A274143/b274143.txt">Table of n, a(n) for n = 0..8805</a>

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

%p A274143 := proc(r)

%p local gs,n,gs2,el,a ;

%p gs := [2,r] ;

%p for n from 3 do

%p gs2 := [] ;

%p for el in gs do

%p gs2 := [op(gs2),el+1,r*el] ;

%p end do:

%p gs := gs2 ;

%p a := 0 ;

%p for el in gs do

%p if type(el,'integer') then

%p a := a+1 :

%p end if;

%p end do:

%p print(n,a) ;

%p end do:

%p end proc:

%p A274143(1/3) ; # _R. J. Mathar_, Jun 17 2016

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

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

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

%Y Cf. A274142.

%K nonn

%O 0,5

%A _Clark Kimberling_, Jun 11 2016

%E More terms from _Kenny Lau_, Jul 04 2016