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

Author

Zhi-Wei Sun, Apr 01 2015

Keywords

• nonn

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.

Examples

			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).
		

Links

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

Crossrefs

Cf. A000217.

Programs

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}]