OFFSET
0,4
COMMENTS
Conjecture: (i) a(n) > 0 for all n. Also, a(n) = 1 only for n = 0, 1, 2, 4, 9, 11, 13, 20, 35, 69, 75, 188.
(ii) For each a = 2,3, any nonnegative integer n can be written as x(x+1)/2 + a*y^2 + z*(3*z+1)/2 with x,y,z nonnegative integers.
Compare part (i) of this conjecture with the conjecture in A160325.
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.
EXAMPLE
a(20) = 1 since 20 = 1*2/2 + 2^2 + 3*(3*3+1)/2.
a(35) = 1 since 35 = 7*8/2 + 0^2 + 2*(3*2+1)/2.
a(69) = 1 since 69 = 2*3/2 + 8^2 + 1*(3*1+1)/2.
a(75) = 1 since 75 = 9*10/2 + 2^2 + 4*(3*4+1)/2.
a(188) = 1 since 188 = 1*2/2 + 0^2 + 11*(3*11+1)/2.
MATHEMATICA
TQ[n_]:=IntegerQ[Sqrt[8n+1]]
Do[r=0; Do[If[TQ[n-4y^2-z(3z+1)/2], r=r+1], {y, 0, Sqrt[n/4]}, {z, 0, (Sqrt[24(n-4y^2)+1]-1)/6}];
Print[n, " ", r]; Label[aa]; Continue, {n, 0, 100}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Feb 04 2015
STATUS
approved