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

%I #17 Oct 02 2017 03:59:28

%S 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 107,130,159,191,234,283,345,418,507,617,747,910,1103,1340,1629,1976,

%U 2402,2914,3542,4300,5223,6344,7701,9359,11361,13801,16761,20353,24725,30021,36468,44285,53788,65328

%N Number of integers in n-th generation of tree T(3^(-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="/A274159/b274159.txt">Table of n, a(n) for n = 0..11841</a>

%e 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 := proc(r)

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

%p gs := [1] ;

%p for n from 2 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 A274159(1/root[3](3)) ; # _R. J. Mathar_, Jun 20 2016

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

%t u = Table[t[[k]] /. x -> 3^(-1/3), {k, 1, z}]; Table[Count[Map[IntegerQ, u[[k]]], True], {k, 1, z}]

%Y Cf. A274142.

%K nonn

%O 0,7

%A _Clark Kimberling_, Jun 12 2016

%E a(15)-a(18) from _R. J. Mathar_, Jun 20 2016

%E More terms from _Kenny Lau_, Jul 04 2016