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A254631
Number of ways to write n as x*(x+1)/2 + y*(3*y+2) + z*(3*z-2) with x,y,z nonnegative integers.
3
1, 2, 1, 1, 1, 1, 3, 2, 2, 2, 1, 3, 1, 1, 2, 2, 4, 2, 2, 2, 2, 3, 3, 3, 2, 1, 3, 5, 2, 3, 1, 2, 2, 2, 5, 1, 5, 4, 2, 2, 3, 5, 3, 3, 4, 4, 3, 3, 2, 3, 2, 3, 3, 2, 3, 5, 4, 5, 3, 2, 5, 4, 6, 2, 2, 3, 6, 3, 3, 4, 3, 7, 3, 4, 3, 2, 4, 4, 4, 6, 3, 3, 4, 4, 4, 5, 5, 4, 3, 2, 3, 5, 8, 3, 3, 3, 7, 3, 3, 8, 4
OFFSET
0,2
COMMENTS
Conjecture: a(n) > 0 for all n, and a(n) > 1 for all n > 35.
We have proved that any nonnegative integer n can be written as x*(x+1)/2 + y*(3*y+2) + z*(3*z-2) with x,y,z integers.
LINKS
Zhi-Wei Sun, On universal sums of polygonal numbers, arXiv:0905.0635 [math.NT], 2009-2015.
EXAMPLE
a(13) = 1 since 13 = 0*1/2 + 1*(3*1+2) + 2*(3*2-2).
a(30) = 1 since 30 = 3*4/2 + 2*(3*2+2) + 2*(3*2-2).
a(35) = 1 since 35 = 1*2/2 + 3*(3*3+2) + 1*(3*1-2).
MATHEMATICA
TQ[n_]:=IntegerQ[Sqrt[8n+1]]
Do[r=0; Do[If[TQ[n-y(3y+2)-z(3z-2)], r=r+1], {y, 0, (Sqrt[3n+1]-1)/3}, {z, 0, (Sqrt[3(n-y(3y+2))+1]+1)/3}];
Print[n, " ", r]; Continue, {n, 0, 100}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Feb 03 2015
STATUS
approved