A274142 - OEIS

%I #27 Jul 04 2016 05:58:04

%S 1,1,1,2,2,4,5,8,11,17,25,37,54,81,119,177,261,388,574,851,1260,1868,

%T 2767,4101,6077,9006,13347,19781,29315,43448,64392,95436,141444,

%U 209636,310705,460501,682519,1011581,1499295,2222155,3293534,4881472,7235018,10723311,15893460,23556367,34913897,51747400

%N Number of integers in n-th generation of tree T(1/2) 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 Guide to related sequences:

%C r           sequence

%C 1/2         A274142

%C 1/3         A274143

%C 1/4         A274144

%C 2/3         A274145

%C 3/4         A274146

%C -1/2        A274147

%C -1/3        A274148

%C -1/4        A274149

%C -2/3        A274150

%C -3/4        A274151

%C 3/2         A274152

%C 5/2         A274153

%C -3/2        A274154

%C -5/2        A274155

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

%C 2^(1/3)     A000930

%C 2^(1/4)     A003269

%C 2^(-1/2)    A274156

%C 3^(-1/2)    A274157

%C 2^(-1/3)    A274158

%C 3^(-1/3)    A274159

%C i           A274160

%C 2i          A206743

%C 3i          A274162

%C 4i          A274163

%C i/2         A274149

%C i/3         A274165

%C i+1         A274166

%C i-1         A274167

%C (-1+3i)/2   A274168

%H Kenny Lau, <a href="/A274142/b274142.txt">Table of n, a(n) for n = 0..5847</a>

%e 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.

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

%t u = Table[t[[k]] /. x -> 1/2, {k, 1, z}];

%t Table[Count[Map[IntegerQ, u[[k]]], True], {k, 1, z}]

%t (* second program: *)

%t 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 *)

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

%K nonn

%O 0,4

%A _Clark Kimberling_, Jun 11 2016

%E More terms from _Jean-François Alcover_, Jun 14 2016

%E More terms from _Kenny Lau_, Jul 04 2016