|
%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
|
