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Tree T generated by these rules: 0 is in T, and if x is in T, then x+1 and -2x are in T, with duplicates deleted as they occur.
2

%I #11 Nov 25 2015 21:22:02

%S 0,1,-2,2,-4,-1,3,4,-8,-6,-3,5,8,-16,-10,-7,-5,6,9,12,16,-32,-24,-18,

%T -15,-12,-9,7,10,13,14,17,20,32,-64,-40,-34,-31,-28,-26,-23,-20,-17,

%U -14,-11,11,15,18,21,24,30,33,36,48,64,-128,-96,-72,-66,-63,-60

%N Tree T generated by these rules: 0 is in T, and if x is in T, then x+1 and -2x are in T, with duplicates deleted as they occur.

%C Every integer occur in T exactly once, so that this is a permutation of the integers. Let g(0) = {0}, g(1) = {1}, g(2) = {-2,2}, g(3) = {-4,-1,3,4}, etc. The number |g(n)| of numbers in the n-th generation is a Fibonacci number except for g(3); see A264800.

%H Clark Kimberling, <a href="/A264799/b264799.txt">Table of n, a(n) for n = 0..10000</a>

%e Generations begin with

%e 0

%e 1

%e -2 2

%e -4 -1 3 4

%e -8 -6 -3 5 8

%e -16 -10 -7 -5 6 9 12 16

%t z = 10; t = Expand[NestList[DeleteDuplicates[Flatten[Map[{# + 1, -2*#} &, #], 1]] &, {0}, z]]; s[0] = t[[1]]; s[n_] := s[n] = Union[t[[n]], s[n - 1]];

%t g[n_] := Complement[s[n], s[n - 1]]; g[1] = {0};

%t Table[Length[g[k]], {k, 1, z}] (* A264800 *)

%t u = Table[g[k], {k, 1, z}]

%t Flatten[u] (* A264799 *)

%Y Cf. A000045, A226080, A264800.

%K sign,tabf,easy

%O 0,3

%A _Clark Kimberling_, Nov 25 2015