OFFSET
0,4
COMMENTS
Conjecture 1: a(n) > 0 for all n >= 0, and a(n) = 1 only for n = 0, 1, 2, 4, 7, 9, 11, 14, 23, 25, 28, 37.
Conjecture 2: Each n = 0,1,2,... can be written as w*(4w+2) + x*(4x-1) + y*(4y-2) + z*(4z-3) with w,x,y,z nonnegative integers.
Conjecture 3: Each n = 0,1,2,... can be written as 4*w^2 + x*(4x+1) + y*(4y-2) + z*(4z-3) with w,x,y,z nonnegative integers.
We have verified that a(n) > 0 for all n = 0..2*10^6. By Theorem 1.3 in the linked 2017 paper of the author, any nonnegative integer can be written as x*(4x-1) + y*(4y-2) + z*(4z-3) with x,y,z integers.
LINKS
Zhi-Wei Sun, Table of n, a(n) for n = 0..10000
Zhi-Wei Sun, A result similar to Lagrange's theorem, J. Number Theory 162(2016), 190-211.
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(11) = 1 with 11 = 1*(4*1+1) + 1*(4*1-1) + 1*(4*1-2) + 1*(4*1-3).
a(23) = 1 with 23 = 2*(4*2+1) + 1*(4*1-1) + 1*(4*1-2) + 0*(4*0-3).
a(25) = 1 with 25 = 0*(4*0+1) + 1*(4*1-1) + 2*(4*2-2) + 2*(4*2-3).
a(28) = 1 with 28 = 2*(4*2+1) + 0*(4*0-1) + 0*(4*0-2) + 2*(4*2-3).
a(37) = 1 with 37 = 1*(4*1+1) + 1*(4*1-1) + 1*(4*1-2) + 3*(4*3-3).
MATHEMATICA
QQ[n_]:=QQ[n]=IntegerQ[Sqrt[16n+1]]&&Mod[Sqrt[16n+1], 8]==1;
tab={}; Do[r=0; Do[If[QQ[n-x(4x-1)-y(4y-2)-z(4z-3)], r=r+1], {x, 0, (Sqrt[16n+1]+1)/8}, {y, 0, (Sqrt[4(n-x(4x-1))+1]+1)/4}, {z, 0, (Sqrt[16(n-x(4x-1)-y(4y-2))+9]+3)/8}]; tab=Append[tab, r], {n, 0, 100}]; Print[tab]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Feb 01 2019
STATUS
approved