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

%I #14 Jul 02 2016 01:25:09

%S 1,1,1,1,2,2,3,4,6,7,10,14,19,25,35,47,64,87,119,161,220,300,407,554,

%T 757,1028,1399,1908,2598,3534,4816,6560,8929,12161,16567,22556,30718,

%U 41843,56981,77597,105693,143944,196029,266991,363634,495228,674481,918629,1251106,1703941,2320726,3160713,4304733

%N Number of integers in n-th generation of tree T(2^(-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 See A274142 for a guide to related sequences.

%H Kenny Lau, <a href="/A274156/b274156.txt">Table of n, a(n) for n = 0..7448</a>

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

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

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

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

%Y Cf. A274142.

%K nonn,easy

%O 0,5

%A _Clark Kimberling_, Jun 12 2016

%E More terms from _Kenny Lau_, Jul 01 2016