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A352286
Number of ways to write n as w + x^2 + 2*y^2 + 3*z^2 + x*y*z, where w is 0 or 1, and x,y,z are nonnegative integers.
4
1, 2, 2, 3, 4, 3, 2, 3, 3, 3, 2, 3, 6, 4, 3, 2, 2, 5, 5, 5, 4, 3, 4, 2, 1, 5, 5, 4, 6, 5, 3, 3, 4, 5, 4, 5, 7, 5, 4, 5, 4, 3, 3, 3, 4, 3, 3, 5, 6, 7, 6, 5, 7, 6, 4, 4, 4, 7, 5, 4, 4, 3, 7, 5, 4, 5, 5, 8, 6, 2, 2, 2, 4, 6, 4, 6, 10, 11, 6, 2, 5, 7, 7, 7, 8, 5, 3, 3, 2, 4, 4, 7, 7, 4, 6, 6, 4, 7, 8, 7, 7
OFFSET
0,2
COMMENTS
Conjecture 1: a(n) = 0 only for n = 106, 744, 5469, 331269. Thus, for any nonnegative integer n not among 106, 744, 5469 and 331269, either n or n - 1 can be written as x^2 + 2*y^2 + 3*z^2 + x*y*z with x,y,z nonnegative integers.
Conjecture 2: a(n) = 1 only for n = 0, 24, 346, 360, 664, 667, 1725, 2589, 3111, 4906, 5035, 8043, 8709, 16810, 18699, 34539, 39256, 51621, 59019, 62799, 108645, 136167, 562696.
We have verified Conjectures 1 and 2 for n = 0..10^6.
EXAMPLE
a(24) = 1 with 24 = 0 + 4^2 + 2*2^2 + 3*0^2 + 4*2*0.
a(346) = 1 with 346 = 1 + 15^2 + 2*3^2 + 3*2^2 + 15*3*2.
a(360) = 1 with 360 = 1 + 9^2 + 2*5^2 + 3*4^2 + 9*5*4.
a(62799) = 1 with 62799 = 1 + 16^2 + 2*169^2 + 3*2^2 + 16*169*2.
a(108645) = 1 with 108645 = 0 + 95^2 + 2*163^2 + 3*3^2 + 95*163*3.
a(136167) = 1 with 136167 = 0 + 2^2 + 2*17^2 + 3*207^2 + 2*17*207.
a(562696) = 1 with 562696 = 0 + 539^2 + 2*20^2 + 3*25^2 + 539*20*25.
MATHEMATICA
SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
tab={}; Do[r=0; Do[If[SQ[4(n-w-2y^2-3z^2)+y^2*z^2], r=r+1], {w, 0, Min[1, n]}, {z, 0, Sqrt[(n-w)/3]}, {y, 0, Sqrt[(n-w-3z^2)/2]}]; tab=Append[tab, r], {n, 0, 100}]; Print[tab]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Mar 10 2022
STATUS
approved