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Number of ordered ways to write n as x*(3x+2) + y*(5y+1)/2 - z^4, where x and y are integers, and z is a nonnegative integer with z^4 <= n.
2

%I #11 Jul 21 2023 09:15:28

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

%T 2,3,2,2,4,3,3,5,9,6,3,4,2,2,2,6,3,3,2,2,3,2,2,4,5,4,5,3,2,2,6,7,4,4,

%U 2,2,4,3,3,3,2,3,4,4,4,2,5

%N Number of ordered ways to write n as x*(3x+2) + y*(5y+1)/2 - z^4, where x and y are integers, and z is a nonnegative integer with z^4 <= n.

%C Conjecture: a(n) > 0 for all n > 0, and a(n) = 1 only for n = 0, 5, 6, 12, 13, 14, 112, 193, 194, 200, 242, 333, 345, 376, 492, 528, 550, 551, 613, 797, 1178, 1195, 1222, 1663, 3380, 3635, 6508, 8755, 9132, 12434, 20087.

%C Compare this conjecture with the conjecture in A270566.

%H Zhi-Wei Sun, <a href="/A270921/b270921.txt">Table of n, a(n) for n = 0..10000</a>

%H Zhi-Wei Sun, <a href="http://dx.doi.org/10.4064/aa127-2-1">Mixed sums of squares and triangular numbers</a>, Acta Arith. 127(2007), 103-113.

%H Zhi-Wei Sun, <a href="https://doi.org/10.1007/s11425-015-4994-4">On universal sums of polygonal numbers</a>, Sci. China Math. 58(2015), no. 7, 1367-1396.

%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1502.03056">On universal sums ax^2+by^2+f(z), aT_x+bT_y+f(z) and zT_x+by^2+f(z)</a>, preprint, arXiv:1502.03056 [math.NT], 2015.

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

%e a(6) = 1 since 6 = 1*(3*1+2) + (-1)*(5*(-1)+1)/2 - 1^4.

%e a(13) = 1 since 13 = 1*(3*1+2) + (-2)*(5*(-2)+1)/2 - 1^4.

%e a(376) = 1 since 376 = 0*(3*0+2) + (-16)*(5*(-16)+1)/2 - 4^4.

%e a(9132) = 1 since 9132 = (-13)*(3*(-13)+2) + 59*(5*59+1)/2 - 3^4.

%e a(12434) = 1 since 12434 = (-21)*(3*(-21)+2) + 78*(5*78+1)/2 - 8^4.

%e a(20087) = 1 since 20087 = 19*(3*19+2) + 87*(5*87+1)/2 - 0^4.

%e 5, 6, 12, 13, 14, 112, 193, 194, 200, 242, 333, 345, 376, 492, 528, 550, 551, 613, 797, 1178, 1195, 1222, 1663, 3380, 3635, 6508, 8755, 9132, 12434, 20087

%t pQ[n_]:=pQ[n]=IntegerQ[Sqrt[40n+1]]&&(Mod[Sqrt[40n+1],10]==1||Mod[Sqrt[40n+1],10]==9)

%t Do[r=0;Do[If[pQ[n+x^4-y(3y+2)],r=r+1],{x,0,n^(1/4)},{y,-Floor[(Sqrt[3(n+x^4)+1]+1)/3],(Sqrt[3(n+x^4)+1]-1)/3}];Print[n," ",r];Continue,{n,0,80}]

%Y Cf. A000583, A001082, A262813, A262815, A262816, A262827, A270469, A270488, A270516, A270533, A270559, A270566, A270920.

%K nonn

%O 0,2

%A _Zhi-Wei Sun_, Mar 25 2016