A256544 - OEIS

%I #7 Apr 01 2015 15:49:12

%S 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,

%T 10,10,9,14,10,11,11,13,9,14,10,12,15,11,13,12,14,12,12,13,15,14,14,

%U 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

%N 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.

%C 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.

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

%H Zhi-Wei Sun, <a href="/A256544/b256544.txt">Table of n, a(n) for n = 0..10000</a>

%e 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).

%t S[n_]:=Union[Table[Floor[x*(x+1)/6], {x, 0, (Sqrt[24n+21]-1)/2}]]

%t L[n_]:=Length[S[n]]

%t 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]];

%t If[MemberQ[S[n], n-Part[S[n],x]-Part[S[n],y]]==True,r=r+1];

%t Continue,{y,x,L[n]}];Label[bb];Continue,{x,1,L[n]}];Label[cc];Print[n," ",r];Continue, {n,0,100}]

%Y Cf. A000217.

%K nonn

%O 0,3

%A _Zhi-Wei Sun_, Apr 01 2015