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

0,7

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.

Links

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

Example

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

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}]

Crossrefs

Cf. A274142.

Sequence in context: A238210 A120170 A136421 * A239905 A016085 A018122

Adjacent sequences: A274156 A274157 A274158 * A274160 A274161 A274162

Keyword

nonn

Author

Clark Kimberling, Jun 12 2016

Extensions

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

More terms from Kenny Lau, Jul 04 2016

Status

approved