A306242 Number of ways to write n as x*(3x+1) + y*(3y-1) + z*(3z+2) + w*(3w-2), where x,y,z,w are nonnegative integers.

1, 1, 1, 1, 1, 2, 2, 2, 2, 1, 3, 2, 2, 1, 3, 4, 3, 3, 2, 4, 3, 3, 4, 3, 4, 3, 3, 4, 3, 4, 7, 4, 5, 4, 3, 4, 4, 6, 6, 3, 9, 6, 2, 5, 5, 8, 4, 6, 7, 6, 5, 6, 3, 5, 9, 6, 8, 7, 8, 7, 7, 8, 7, 4, 9, 8, 6, 6, 7, 7, 13, 9, 6, 6, 7, 11, 4, 6, 11, 9, 12

Offset

0,6

Comments

Conjecture: a(n) > 0 for all n >= 0, and a(n) = 1 only for n = 0, 1, 2, 3, 4, 9, 13. Moreover, any nonnegative integer n can be written as x*(3x+1) + y*(3y-1) + z*(3z+2) + w*(3w-2), where x,y,z,w are nonnegative integers with x or y even.

The conjecture has been verified for n up to 10^6.

By Theorem 1.3 of the linked 2017 paper of the author, each nonnegative integer can be written as x*(3x+1) + y*(3y-1) + z*(3z+2) + 0*(3*0-2) with x,y,z integers.

We also have some other similar conjectures. For example, we conjecture that every n = 0,1,2,... can be written as x*(5x+1)/2 + y*(5y-1)/2 + z*(5z+3)/2 + w*(5w-3)/2 with x,y,z,w nonnegative 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(1) = 1 with 1 = 0*(3*0+1) + 0*(3*0-1) + 0*(3*0+2) + 1*(3*1-2).
a(3) = 1 with 3 = 0*(3*0+1) + 1*(3*1-1) + 0*(3*0+2) + 1*(3*1-2).
a(4) = 1 with 4 = 1*(3*1+1) + 0*(3*0-1) + 0*(3*0+2) + 0*(3*0-2).
a(9) = 1 with 9 = 1*(3*1+1) + 0*(3*0-1) + 1*(3*1+2) + 0*(3*0-2).
a(13) = 1 with 13 = 0*(3*0+1) + 0*(3*0-1) + 1*(3*1+2) + 2*(3*2-2).

Mathematica

OctQ[n_]:=OctQ[n]=IntegerQ[Sqrt[3n+1]]&&(n==0||Mod[Sqrt[3n+1]+1, 3]==0);
tab={}; Do[r=0; Do[If[OctQ[n-x(3x+2)-y(3y+1)-z(3z-1)], r=r+1], {x, 0, (Sqrt[3n+1]-1)/3}, {y, 0, (Sqrt[12(n-x(3x+2))+1]-1)/6}, {z, 0, (Sqrt[12(n-x(3x+2)-y(3y+1))+1]+1)/6}]; tab=Append[tab, r], {n, 0, 80}]; Print[tab]

Crossrefs

Cf. A000567, A045944, A049450, A049451, A255350.

Sequence in context: A145443 A192056 A207385 * A118144 A136691 A361199

Adjacent sequences: A306239 A306240 A306241 * A306243 A306244 A306245

Keyword

nonn

Author

Zhi-Wei Sun, Jan 31 2019

Status

approved