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
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Apr 01 2015
STATUS
approved