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A256910 Trace of the enhanced triangular-number representation of n.

%I A256910 #6 Apr 14 2015 11:05:48
%S A256910 0,1,2,3,1,2,6,1,2,3,10,1,2,3,1,15,1,2,3,1,2,21,1,2,3,1,2,6,28,1,2,3,
%T A256910 1,2,6,1,36,1,2,3,1,2,6,1,2,45,1,2,3,1,2,6,1,2,3,55,1,2,3,1,2,6,1,2,3,
%U A256910 10,66,1,2,3,1,2,6,1,2,3,10,1,78,1,2,3,1
%N A256910 Trace of the enhanced triangular-number representation of n.
%C A256910 See A256909 for definitions.
%H A256910 Clark Kimberling, <a href="/A256910/b256910.txt">Table of n, a(n) for n = 0..1000</a>
%e A256910 R(0) = 0, trace = 0;
%e A256910 R(1) = 1, trace = 1;
%e A256910 R(2) = 2, trace = 2;
%e A256910 R(3) = 3, trace = 3;
%e A256910 R(4) = 3 + 1, trace = 1;
%e A256910 R(5) = 3 + 2, trace = 2;
%e A256910 R(6) = 6, trace = 6;
%e A256910 R(119) = 105 + 10 + 3 + 1, trace = 1.
%t A256910 b[n_] := n (n + 1)/2; bb = Insert[Table[b[n], {n, 0, 200}], 2, 3]
%t A256910 s[n_] := Table[b[n], {k, 1, n + 1}];
%t A256910 h[1] = {0, 1, 2}; h[n_] := Join[h[n - 1], s[n]];
%t A256910 g = h[200]; r[0] = {0};
%t A256910 r[n_] := If[MemberQ[bb, n], {n}, Join[{g[[n]]}, r[n - g[[n]]]]];
%t A256910 t = Table[r[n], {n, 0, 120}] (*A256909 before concatenation*)
%t A256910 Flatten[t]  (*A256909*)
%t A256910 Table[Last[r[n]], {n, 0, 120}]    (*A256910*)
%t A256910 Table[Length[r[n]], {n, 0, 120}]  (*A256911*)
%Y A256910 Cf. A000217, A256909 (definitions), A256911 (number of terms), A255974 (minimal alternating triangular-number representations).
%K A256910 nonn,easy
%O A256910 0,3
%A A256910 _Clark Kimberling_, Apr 13 2015