A256909 -id:A256909 - cp's OEIS frontend

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

Clark Kimberling, Apr 13 2015

See A256909 for definitions.

			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.

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

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

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*)

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

Original entry on oeis.org

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

Clark Kimberling, Apr 14 2015

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

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

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

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

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*)

cp's OEIS Frontend

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

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