A256698 -id:A256698 - cp's OEIS frontend

A256697 Trace of the minimal alternating triangular-number representation of n.

0, 1, -1, 3, 1, -1, 6, -3, 1, -1, 10, -1, -3, 1, -1, 15, 1, -1, -3, 1, -1, 21, -6, 1, -1, -3, 1, -1, 28, 3, -6, 1, -1, -3, 1, -1, 36, -1, 3, -6, 1, -1, -3, 1, -1, 45, 1, -1, 3, -6, 1, -1, -3, 1, -1, 55, -10, 1, -1, 3, -6, 1, -1, -3, 1, -1, 66, 1, -10, 1, -1

Offset: 0

Clark Kimberling, Apr 11 2015

See A255974 for definitions.

			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(8) = 10 - 3 + 1; trace = 1
R(11) = 15 - 6 + 3 - 1; trace = -1

Clark Kimberling, Table of n, a(n) for n = 0..1000

Cf. A255974, A256698, A256699.

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]]]
t = Table[r[n], {n, 0, 120}]  (* A255974 representations *)
t = Table[Last[r[n]], {n, 0, 120}] (* A256697 *)

A256699 Numbers with negative triangular trace.

Original entry on oeis.org

2, 5, 7, 9, 11, 12, 14, 17, 18, 20, 22, 24, 25, 27, 30, 32, 33, 35, 37, 39, 41, 42, 44, 47, 49, 51, 52, 54, 56, 58, 60, 62, 63, 65, 68, 70, 72, 74, 75, 77, 81, 83, 85, 87, 88, 90, 92, 95, 97, 99, 101, 102, 104, 107, 110, 112, 114, 116, 117, 119, 121, 123

Offset: 1

Clark Kimberling, Apr 11 2015

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

			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.

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A255974, A256697, A256698 (complement)

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 *)

cp's OEIS Frontend

A256697 Trace of the minimal alternating triangular-number representation of n.

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