A256910 Trace of the enhanced triangular-number representation of n.

0, 1, 2, 3, 1, 2, 6, 1, 2, 3, 10, 1, 2, 3, 1, 15, 1, 2, 3, 1, 2, 21, 1, 2, 3, 1, 2, 6, 28, 1, 2, 3, 1, 2, 6, 1, 36, 1, 2, 3, 1, 2, 6, 1, 2, 45, 1, 2, 3, 1, 2, 6, 1, 2, 3, 55, 1, 2, 3, 1, 2, 6, 1, 2, 3, 10, 66, 1, 2, 3, 1, 2, 6, 1, 2, 3, 10, 1, 78, 1, 2, 3, 1

Offset

0,3

Comments

See A256909 for definitions.

Links

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

Example

R(0) = 0, trace = 0;
R(1) = 1, trace = 1;
R(2) = 2, trace = 2;
R(3) = 3, trace = 3;
R(4) = 3 + 1, trace = 1;
R(5) = 3 + 2, trace = 2;
R(6) = 6, trace = 6;
R(119) = 105 + 10 + 3 + 1, trace = 1.

Mathematica

b[n_] := n (n + 1)/2; bb = Insert[Table[b[n], {n, 0, 200}], 2, 3]
s[n_] := Table[b[n], {k, 1, n + 1}];
h[1] = {0, 1, 2}; h[n_] := Join[h[n - 1], s[n]];
g = h[200]; r[0] = {0};
r[n_] := If[MemberQ[bb, n], {n}, Join[{g[[n]]}, r[n - g[[n]]]]];
t = Table[r[n], {n, 0, 120}] (*A256909 before concatenation*)
Flatten[t]  (*A256909*)
Table[Last[r[n]], {n, 0, 120}]    (*A256910*)
Table[Length[r[n]], {n, 0, 120}]  (*A256911*)

Crossrefs

Cf. A000217, A256909 (definitions), A256911 (number of terms), A255974 (minimal alternating triangular-number representations).

Sequence in context: A117488 A307668 A308173 * A181176 A131108 A128255

Adjacent sequences: A256907 A256908 A256909 * A256911 A256912 A256913

Keyword

nonn,easy

Author

Clark Kimberling, Apr 13 2015

Status

approved