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%I #7 Mar 10 2015 01:56:11
%S 1,1,2,1,3,1,2,3,1,2,4,1,3,4,1,3,5,1,2,3,4,1,2,3,5,1,2,4,5,1,2,4,6,1,
%T 3,4,5,1,3,4,6,1,3,5,6,1,3,5,7,1,2,3,4,5,1,2,3,4,6,1,2,3,5,6,1,2,3,5,
%U 7,1,2,4,5,6,1,2,4,5,7,1,2,4,6,7,1,2
%N Infinite tree (flattened) generated as follows: generation g(1) = (1); thereafter, putting h = 2^n, each (1,x(2),...,x(h)) in generation g(n) has 1st and 2nd offspring, namely (1,x(2),...,x(h),x(h)+1) and (1,x(2),...,x(h),x(h)+2).
%C Generation n consists of 2^(n-1) increasing n-tuples that have maximal gapsize 2.
%H Clark Kimberling, <a href="/A255809/b255809.txt">Table of n, a(n) for n = 1..4000</a>
%e generation g(1) = (1);
%e g(2) = (1,2), (1,3);
%e g(3) = (1,2,3), (1,2,4), (1,3,4), (1,3,5);
%e g(4) = (1,2,3,4), (1,2,3,5), (1,2,4,5), (1,2,4,6), (1,3,4,5), (1,3,4,6), (1,3,5,6), (1,3,5,7).
%t z = 5; t[n_] := t[n] = Join[{{First[#]}}, Rest[#]] &[Sort[Flatten[NestList[Map[Flatten, Transpose[Map[Flatten[#, 1] &, {{#, #}, {1 + Map[Last, #], 2 + Map[Last, #]}}]]] &, 1(*seed*), #], 1]] &[n(*iterations*)]]
%t Column[Table[t[n], {n, 1, z}]] (* 1st z generations *)
%t Flatten[t[6]] (* A255809, _Peter J. C. Moses_, Mar 05 2015 *)
%Y Cf. A255810.
%K nonn,easy
%O 1,3
%A _Clark Kimberling_, Mar 09 2015
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