A274143 Number of integers in n-th generation of tree T(1/3) defined in Comments.

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, 356, 462, 597, 780, 1010, 1311, 1706, 2210, 2873, 3732, 4841, 6294, 8168, 10608, 13781, 17886, 23237, 30172, 39177, 50891, 66072, 85813, 111446, 144706, 187947, 244059, 316937, 411618, 534503, 694153, 901461

Offset: 0

Author

Clark Kimberling, Jun 11 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

			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.
		

Links

• Kenny Lau, Table of n, a(n) for n = 0..8805

Crossrefs

Cf. A274142.

Programs

Maple

A274143 := proc(r)
    local gs,n,gs2,el,a ;
    gs := [2,r] ;
    for n from 3 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:
A274143(1/3) ; # R. J. Mathar, Jun 17 2016

Mathematica

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

Extensions

More terms from Kenny Lau, Jul 04 2016