A274142 Number of integers in n-th generation of tree T(1/2) defined in Comments.

1, 1, 1, 2, 2, 4, 5, 8, 11, 17, 25, 37, 54, 81, 119, 177, 261, 388, 574, 851, 1260, 1868, 2767, 4101, 6077, 9006, 13347, 19781, 29315, 43448, 64392, 95436, 141444, 209636, 310705, 460501, 682519, 1011581, 1499295, 2222155, 3293534, 4881472, 7235018, 10723311, 15893460, 23556367, 34913897, 51747400

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.

Guide to related sequences:

r sequence

1/2 A274142

1/3 A274143

1/4 A274144

2/3 A274145

3/4 A274146

-1/2 A274147

-1/3 A274148

-1/4 A274149

-2/3 A274150

-3/4 A274151

3/2 A274152

5/2 A274153

-3/2 A274154

-5/2 A274155

2^(1/2) A000045 (Fibonacci numbers)

2^(1/3) A000930

2^(1/4) A003269

2^(-1/2) A274156

3^(-1/2) A274157

2^(-1/3) A274158

3^(-1/3) A274159

i A274160

2i A206743

3i A274162

4i A274163

i/2 A274149

i/3 A274165

i+1 A274166

i-1 A274167

(-1+3i)/2 A274168

Examples

			If r = 1/2, then g(3) = {3,2r,r+1, r^2}, in which the integers are 3 and 1, so that a(3) = 2.
		

Links

• Kenny Lau, Table of n, a(n) for n = 0..5847

Crossrefs

Cf. A274143-A274160, A274162, A274163, A274165-A274168.

Programs

Mathematica

z = 18; t = Join[{{0}}, Expand[NestList[DeleteDuplicates[Flatten[Map[{# + 1, x*#} &, #], 1]] &, {1}, z]]];
u = Table[t[[k]] /. x -> 1/2, {k, 1, z}];
Table[Count[Map[IntegerQ, u[[k]]], True], {k, 1, z}]
(* second program: *)
T[0] = {0}; T[n_] := T[n] = Complement[Join[T[n-1]+1, x*T[n-1]], T[n-1]]; Reap[For[n = 0, n <= 25, n++, cnt = Count[T[n] /. x -> 1/2, Integer]; Print[n, " ", cnt]; Sow[cnt]]][[2, 1]] (* _Jean-François Alcover, Jun 14 2016 *)

Extensions

More terms from Jean-François Alcover, Jun 14 2016

More terms from Kenny Lau, Jul 04 2016