A256911 Number of terms in the enhanced triangular-number representation of n.

1, 1, 1, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 3, 1, 2, 2, 2, 3, 3, 1, 2, 2, 2, 3, 3, 2, 1, 2, 2, 2, 3, 3, 2, 3, 1, 2, 2, 2, 3, 3, 2, 3, 3, 1, 2, 2, 2, 3, 3, 2, 3, 3, 3, 1, 2, 2, 2, 3, 3, 2, 3, 3, 3, 2, 1, 2, 2, 2, 3, 3, 2, 3, 3, 3, 2, 3, 1, 2, 2, 2, 3, 3, 2, 3

Offset: 0

Author

Clark Kimberling, Apr 14 2015

Keywords

• nonn
• easy

Comments

See A256909 for definitions. The least n for which R(n) has 5 terms is given by R(7259) = 7140 + 105 + 10 + 3 + 1.

Examples

			R(4) = 3 + 1, so a(4) = 2.
R(119) = 105 + 10 + 3 + 1, so a(119) = 4.
		

Links

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

Crossrefs

Cf. A000217, A256909 (representations) A256910 (trace).

Programs

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*)