A256544 Number of ways to write n as the sum of three unordered elements of the set {floor(T(x)/3): x = 1,2,3,...}, where T(x) denotes the triangular number x*(x+1)/2.

1, 1, 2, 3, 3, 4, 4, 5, 4, 6, 5, 6, 6, 6, 6, 8, 6, 8, 7, 9, 7, 9, 8, 8, 9, 9, 9, 10, 9, 9, 11, 9, 12, 10, 10, 9, 14, 10, 11, 11, 13, 9, 14, 10, 12, 15, 11, 13, 12, 14, 12, 12, 13, 15, 14, 14, 11, 16, 11, 17, 14, 14, 14, 16, 13, 16, 15, 17, 12, 15, 17, 15, 17, 15, 14, 20, 13, 15, 19, 14, 18, 16, 21, 12, 19, 15, 16, 22, 18, 15, 18, 14, 21, 19, 18, 18, 17, 19, 18, 17, 18

Offset

0,3

Comments

Conjecture: For any positive integer m, every nonnegative integer n can be written as floor(T(x)/m) + floor(T(y)/m) + floor(T(z)/m) with x,y,z nonnegative integers.

In the case m = 1, this is a well-known result in number theory.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 0..10000

Example

a(4) = 3 since 4 = floor(T(1)/3) + floor(T(2)/3) + floor(T(4)/3) = floor(T(1)/3) + floor(T(3)/3) + floor(T(3)/3) = floor(T(2)/3) + floor(T(2)/3) + floor(T(3)/3).

Mathematica

S[n_]:=Union[Table[Floor[x*(x+1)/6], {x, 0, (Sqrt[24n+21]-1)/2}]]
L[n_]:=Length[S[n]]
Do[r=0; Do[If[Part[S[n], x]>n/3, Goto[cc]]; Do[If[Part[S[n], x]+2*Part[S[n], y]>n, Goto[bb]];
If[MemberQ[S[n], n-Part[S[n], x]-Part[S[n], y]]==True, r=r+1];
Continue, {y, x, L[n]}]; Label[bb]; Continue, {x, 1, L[n]}]; Label[cc]; Print[n, " ", r]; Continue, {n, 0, 100}]

Crossrefs

Cf. A000217.

Sequence in context: A336751 A294991 A300118 * A321211 A336515 A130500

Adjacent sequences: A256541 A256542 A256543 * A256545 A256546 A256547

Keyword

nonn

Author

Zhi-Wei Sun, Apr 01 2015

Status

approved