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Number of ways to write n as x*(2x-1) + y*(3y-1) + z*(4z-1) + w*(5w-1), where x,y,z are nonnegative integers and w is 0 or 1.
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%I #21 Feb 01 2019 00:34:48

%S 1,1,1,2,2,2,3,2,2,2,3,2,1,2,3,4,2,3,3,3,4,2,2,1,4,3,1,1,5,4,3,3,3,4,

%T 4,3,1,3,3,5,1,2,4,5,4,4,2,3,7,3,3,2,5,3,3,2,2,3,4,5,1,4,6,6,2,3,5,3,

%U 3,3,5,4,5,5,3,6,6,4,3,4,5,2,3,4,4,5,2,2,5,6,6,1,5,3,6,2,4,3,4,4,2

%N Number of ways to write n as x*(2x-1) + y*(3y-1) + z*(4z-1) + w*(5w-1), where x,y,z are nonnegative integers and w is 0 or 1.

%C Conjecture: a(n) > 0 for any nonnegative integer n.

%C This has been verified for n up to 10^6. By Theorem 1.2 of the linked 2017 paper of the author, any nonnegative integer can be written as x*(2x-1) + y*(3y-1) + z*(4z-1) with x,y,z integers.

%C We have some other similar conjectures. For example, we conjecture that each n = 0,1,2,... can be written as x*(3x-1)/2 + y*(5y-1)/2 + z*(7z-1)/2 + w*(9w-1)/2) (or x*(x-1) + y*(2y-1) + z*(3z-1) + w*(4w-1)) with x,y,z,w nonnegative integers.

%H Zhi-Wei Sun, <a href="/A306249/b306249.txt">Table of n, a(n) for n = 0..10000</a>

%H Zhi-Wei Sun, <a href="http://dx.doi.org/10.1016/j.jnt.2015.10.014">A result similar to Lagrange's theorem</a>, J. Number Theory 162(2016), 190-211.

%H Zhi-Wei Sun, <a href="http://dx.doi.org/10.1016/j.jnt.2016.07.024">On x(ax+1)+y(by+1)+z(cz+1) and x(ax+b)+y(ay+c)+z(az+d)</a>, J. Number Theory 171(2017), 275-283.

%e a(1) = 1 with 1 = 1*(2*1-1) + 0*(3*0-1) + 0*(4*0-1) + 0*(5*0-1).

%e a(2) = 1 with 2 = 0*(2*0-1) + 1*(3*1-1) + 0*(4*0-1) + 0*(5*0-1).

%e a(12) = 1 with 12 = 2*(2*2-1) + 1*(3*1-1) + 0*(4*0-1) + 1*(5*1-1).

%e a(26) = 1 with 26 = 2*(2*2-1) + 1*(3*1-1) + 2*(4*2-1) + 1*(5*1-1).

%e a(220) = 1 with 220 = 6*(2*6-1) + 7*(3*7-1) + 2*(4*2-1) + 0*(5*0-1).

%e a(561) = 1 with 561 = 17*(2*17-1) + 0*(3*0-1) + 0*(4*0-1) + 0*(5*0-1).

%e a(1356) = 1 with 1356 = 23*(2*23-1) + 1*(3*1-1) + 9*(4*9-1) + 1*(5*1-1).

%t HexQ[n_]:=HexQ[n]=IntegerQ[Sqrt[8n+1]]&&(n==0||Mod[Sqrt[8n+1]+1,4]==0);

%t tab={};Do[r=0;Do[If[HexQ[n-x(5x-1)-y(4y-1)-z(3z-1)],r=r+1],{x,0,Min[1,(Sqrt[20n+1]+1)/10]},{y,0,(Sqrt[16(n-x(5x-1))+1]+1)/8},{z,0,(Sqrt[12(n-x(5x-1)-y(4y-1))+1]+1)/6}];tab=Append[tab,r],{n,0,100}];Print[tab]

%Y Cf. A000326, A000384, A002378, A033991, A049450, A255350, A306225, A306227, A306239, A306242.

%K nonn

%O 0,4

%A _Zhi-Wei Sun_, Jan 31 2019