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

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, 107, 130, 159, 191, 234, 283, 345, 418, 507, 617, 747, 910, 1103, 1340, 1629, 1976, 2402, 2914, 3542, 4300, 5223, 6344, 7701, 9359, 11361, 13801, 16761, 20353, 24725, 30021, 36468, 44285, 53788, 65328

Offset: 0

Author

Clark Kimberling, Jun 12 2016

Keywords

• nonn

Comments

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.

See A274142 for a guide to related sequences.

Examples

			If r = 3^(-1/3), then g(3) = {3,2r,r+1, r^2}, in which the number of integers is a(3) = 1.
		

Links

• Kenny Lau, Table of n, a(n) for n = 0..11841

Crossrefs

Cf. A274142.

Programs

Maple

A274159 := proc(r)
    local gs, n, gs2, el, a ;
    gs := [1] ;
    for n from 2 do
        gs2 := [] ;
        for el in gs do
            gs2 := [op(gs2), el+1, r*el] ;
        end do:
        gs := gs2 ;
        a := 0 ;
        for el in gs do
            if type(el, 'integer') then
                 a := a+1 :
            end if;
        end do:
        print(n, a) ;
    end do:
end proc:
A274159(1/root[3](3)) ; # R. J. Mathar, Jun 20 2016

Mathematica

z = 18; t = Join[{{0}}, Expand[NestList[DeleteDuplicates[Flatten[Map[{# + 1, x*#} &, #], 1]] &, {1}, z]]];
u = Table[t[[k]] /. x -> 3^(-1/3), {k, 1, z}]; Table[Count[Map[IntegerQ, u[[k]]], True], {k, 1, z}]

Extensions

a(15)-a(18) from R. J. Mathar, Jun 20 2016

More terms from Kenny Lau, Jul 04 2016