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A256698 Numbers with positive triangular trace.

%I A256698 #8 Jul 27 2024 01:26:16
%S A256698 1,3,4,6,8,10,13,15,16,19,21,23,26,28,29,31,34,36,38,40,43,45,46,48,
%T A256698 50,53,55,57,59,61,64,66,67,69,71,73,76,78,79,80,82,84,86,89,91,93,94,
%U A256698 96,98,100,103,105,106,108,109,111,113,115,118,120,122,124
%N A256698 Numbers with positive triangular trace.
%C A256698 Are the differences all 1, 2, or 3?  See A255974 for definitions.
%H A256698 Clark Kimberling, <a href="/A256698/b256698.txt">Table of n, a(n) for n = 1..1000</a>
%e A256698 R(0) = 0; trace = 0
%e A256698 R(1) = 1; trace = 1
%e A256698 R(2) = 3 - 1; trace = -1
%e A256698 R(3) = 3; trace = 3
%e A256698 R(4) = 6 - 3 + 1; trace = 1
%e A256698 R(5) = 6 - 1; trace = -1
%e A256698 R(6) = 6; trace = 6
%e A256698 R(7) = 10 - 3; trace = -3
%e A256698 Thus, 1, 3, 4, 6, ... have positive trace, and 2, 5, 7, .... have negative trace.
%t A256698 b[n_] := n (n + 1)/2; bb = Table[b[n], {n, 0, 1000}];
%t A256698 s[n_] := Table[b[n], {k, 1, n}];
%t A256698 h[1] = {1}; h[n_] := Join[h[n - 1], s[n]]; g = h[100]; r[0] = {0};
%t A256698 r[n_] := If[MemberQ[bb, n], {n}, Join[{g[[n]]}, -r[g[[n]] - n]]]
%t A256698 Table[Last[r[n]], {n, 0, 300}]   (* A256697 *)
%t A256698 Select[Range[200], Last[r[#]] > 0 &]  (* A256698 *)
%t A256698 Select[Range[200], Last[r[#]] < 0 &]  (* A256699 *)
%Y A256698 Cf. A255974, A256697, A256699 (complement).
%K A256698 nonn,easy
%O A256698 1,2
%A A256698 _Clark Kimberling_, Apr 11 2015