A306260 Number of ways to write n as w*(4w+1) + x*(4x-1) + y*(4y-2) + z*(4z-3) with w,x,y,z nonnegative integers.

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

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

Cf. A001107, A002939, A007742, A033991, A255350, A306225, A306227, A306239, A306240, A306249, A306250.

Sequence in context: A237253 A080634 A109925 * A180227 A001468 A014675

Adjacent sequences: A306257 A306258 A306259 * A306261 A306262 A306263

Keyword

nonn

Author

Zhi-Wei Sun, Feb 01 2019

Status

approved