A256698 Numbers with positive triangular trace.
1, 3, 4, 6, 8, 10, 13, 15, 16, 19, 21, 23, 26, 28, 29, 31, 34, 36, 38, 40, 43, 45, 46, 48, 50, 53, 55, 57, 59, 61, 64, 66, 67, 69, 71, 73, 76, 78, 79, 80, 82, 84, 86, 89, 91, 93, 94, 96, 98, 100, 103, 105, 106, 108, 109, 111, 113, 115, 118, 120, 122, 124
Offset: 1
Examples
R(0) = 0; trace = 0 R(1) = 1; trace = 1 R(2) = 3 - 1; trace = -1 R(3) = 3; trace = 3 R(4) = 6 - 3 + 1; trace = 1 R(5) = 6 - 1; trace = -1 R(6) = 6; trace = 6 R(7) = 10 - 3; trace = -3 Thus, 1, 3, 4, 6, ... have positive trace, and 2, 5, 7, .... have negative trace.
Links
- Clark Kimberling, Table of n, a(n) for n = 1..1000
Programs
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Mathematica
b[n_] := n (n + 1)/2; bb = Table[b[n], {n, 0, 1000}]; s[n_] := Table[b[n], {k, 1, n}]; h[1] = {1}; h[n_] := Join[h[n - 1], s[n]]; g = h[100]; r[0] = {0}; r[n_] := If[MemberQ[bb, n], {n}, Join[{g[[n]]}, -r[g[[n]] - n]]] Table[Last[r[n]], {n, 0, 300}] (* A256697 *) Select[Range[200], Last[r[#]] > 0 &] (* A256698 *) Select[Range[200], Last[r[#]] < 0 &] (* A256699 *)
Comments