OFFSET
0,2
COMMENTS
Conjecture: a(n) > 0 for all n. Also, a(n) = 1 only for n = 0, 5, 13, 14, 20, 112, 125.
Compare this conjecture with the conjecture in A160324.
The conjecture that a(n) > 0 for all n = 0,1,2,... appeared in Conjecture 1.2(ii) of the author's JNT paper in the links. - Zhi-Wei Sun, Oct 03 2016
LINKS
Zhi-Wei Sun, Table of n, a(n) for n = 0..10000
Zhi-Wei Sun, On universal sums of polygonal numbers, arXiv:0905.0635 [math.NT], 2009-2015.
Zhi-Wei Sun, On x(ax+1)+y(by+1)+z(cz+1) and x(ax+b)+y(ay+c)+z(az+d), J. Number Theory 171(2017), 275-283.
EXAMPLE
a(20) = 1 since 20 = 2^2 + 3*(3*3+1)/2 + 1*(2*1-1).
a(112) = 1 since 112 = 7^2 + 6*(3*6+1)/2 + 2*(2*2-1).
a(125) = 1 since 125 = 5^2 + 8*(3*8+1)/2 + 0*(2*0-1).
MATHEMATICA
SQ[n_]:=IntegerQ[Sqrt[n]]
Do[r=0; Do[If[SQ[n-y(3y+1)/2-z(2z-1)], r=r+1], {y, 0, (Sqrt[24n+1]-1)/6}, {z, 0, (Sqrt[8(n-y(3y+1)/2)+1]+1)/4}];
Print[n, " ", r]; Continue, {n, 0, 100}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Feb 04 2015
STATUS
approved