A256699 - OEIS

%I #8 Jul 27 2024 01:26:12

%S 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 49,51,52,54,56,58,60,62,63,65,68,70,72,74,75,77,81,83,85,87,88,90,92,

%U 95,97,99,101,102,104,107,110,112,114,116,117,119,121,123

%N Numbers with negative triangular trace.

%C Are the differences all 1, 2, 3, or 4?  See A255974 for definitions.

%H Clark Kimberling, <a href="/A256699/b256699.txt">Table of n, a(n) for n = 1..1000</a>

%e R(0) = 0; trace = 0

%e R(1) = 1; trace = 1

%e R(2) = 3 - 1; trace = -1

%e R(3) = 3; trace = 3

%e R(4) = 6 - 3 + 1; trace = 1

%e R(5) = 6 - 1; trace = -1

%e R(6) = 6; trace = 6

%e R(7) = 10 - 3; trace = -3

%e Thus, 1, 3, 4, 6, ... have positive trace, and 2, 5, 7, .... have negative trace.

%t b[n_] := n (n + 1)/2; bb = Table[b[n], {n, 0, 1000}];

%t s[n_] := Table[b[n], {k, 1, n}];

%t h[1] = {1}; h[n_] := Join[h[n - 1], s[n]]; g = h[100]; r[0] = {0};

%t r[n_] := If[MemberQ[bb, n], {n}, Join[{g[[n]]}, -r[g[[n]] - n]]]

%t Table[Last[r[n]], {n, 0, 300}]   (* A256697 *)

%t Select[Range[200], Last[r[#]] > 0 &]  (* A256698 *)

%t Select[Range[200], Last[r[#]] < 0 &]  (* A256699 *)

%Y Cf. A255974, A256697, A256698 (complement)

%K nonn,easy

%O 1,1

%A _Clark Kimberling_, Apr 11 2015