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A256699 Numbers with negative triangular trace.

%I A256699 #8 Jul 27 2024 01:26:12
%S A256699 2,5,7,9,11,12,14,17,18,20,22,24,25,27,30,32,33,35,37,39,41,42,44,47,
%T A256699 49,51,52,54,56,58,60,62,63,65,68,70,72,74,75,77,81,83,85,87,88,90,92,
%U A256699 95,97,99,101,102,104,107,110,112,114,116,117,119,121,123
%N A256699 Numbers with negative triangular trace.
%C A256699 Are the differences all 1, 2, 3, or 4?  See A255974 for definitions.
%H A256699 Clark Kimberling, <a href="/A256699/b256699.txt">Table of n, a(n) for n = 1..1000</a>
%e A256699 R(0) = 0; trace = 0
%e A256699 R(1) = 1; trace = 1
%e A256699 R(2) = 3 - 1; trace = -1
%e A256699 R(3) = 3; trace = 3
%e A256699 R(4) = 6 - 3 + 1; trace = 1
%e A256699 R(5) = 6 - 1; trace = -1
%e A256699 R(6) = 6; trace = 6
%e A256699 R(7) = 10 - 3; trace = -3
%e A256699 Thus, 1, 3, 4, 6, ... have positive trace, and 2, 5, 7, .... have negative trace.
%t A256699 b[n_] := n (n + 1)/2; bb = Table[b[n], {n, 0, 1000}];
%t A256699 s[n_] := Table[b[n], {k, 1, n}];
%t A256699 h[1] = {1}; h[n_] := Join[h[n - 1], s[n]]; g = h[100]; r[0] = {0};
%t A256699 r[n_] := If[MemberQ[bb, n], {n}, Join[{g[[n]]}, -r[g[[n]] - n]]]
%t A256699 Table[Last[r[n]], {n, 0, 300}]   (* A256697 *)
%t A256699 Select[Range[200], Last[r[#]] > 0 &]  (* A256698 *)
%t A256699 Select[Range[200], Last[r[#]] < 0 &]  (* A256699 *)
%Y A256699 Cf. A255974, A256697, A256698 (complement)
%K A256699 nonn,easy
%O A256699 1,1
%A A256699 _Clark Kimberling_, Apr 11 2015