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A350012 Number of ways to write n as 4*x^4 + y^2 + (z^2 + 4^w)/2 with x,y,z,w nonnegative integers.

%I A350012 #15 Dec 09 2021 08:16:29
%S A350012 1,2,1,1,4,4,1,3,5,5,3,3,4,7,3,2,6,5,2,4,6,2,2,5,4,6,2,2,6,7,2,2,6,5,
%T A350012 5,4,3,7,5,5,8,6,2,6,9,4,2,4,5,8,3,3,5,8,3,6,5,3,6,4,6,5,6,1,10,9,2,6,
%U A350012 11,8,1,7,5,11,6,4,7,10,3,6,10,4,8,8,6,8,6,5,11,13,5,1,11,8,3,4,4,9,7,6
%N A350012 Number of ways to write n as 4*x^4 + y^2 + (z^2 + 4^w)/2 with x,y,z,w nonnegative integers.
%C A350012 Conjecture: a(n) > 0 for all n > 0.
%C A350012 This is a new refinement of Lagrange's four-square theorem since (x^2 + y^2)/2 = ((x+y)/2)^2 + ((x-y)/2)^2. We have verified the conjecture for n up to 10^6.
%C A350012 See also A349661 for a similar conjecture.
%C A350012 We also have some other conjectures of such a type.
%H A350012 Zhi-Wei Sun, <a href="/A350012/b350012.txt">Table of n, a(n) for n = 1..10000</a>
%H A350012 Zhi-Wei Sun, <a href="http://dx.doi.org/10.1016/j.jnt.2016.11.008">Refining Lagrange's four-square theorem</a>, J. Number Theory 175(2017), 167--190.
%H A350012 Zhi-Wei Sun, <a href="https://doi.org/10.1142/S1793042119501045">Restricted sums of four squares</a>, Int. J. Number Theory 15(2019), 1863-1893.  See also <a href="http://arxiv.org/abs/1701.05868">arXiv:1701.05868 [math.NT]</a>.
%H A350012 Zhi-Wei Sun, <a href="http://hitpress.hit.edu.cn/2021/1015/c12593a261001/page.htm">New Conjectures in Number Theory and Combinatorics</a> (in Chinese), Harbin Institute of Technology Press, 2021.
%e A350012 a(1) = 4*0^4 + 0^2 + (1^2 + 4^0)/2.
%e A350012 a(3) = 1 with 3 = 4*0^4 + 1^2 + (0^2 + 4)/2.
%e A350012 a(4) = 1 with 4 = 4*0^4 + 0^2 + (2^2 + 4)/2.
%e A350012 a(7) = 1 with 7 = 4*1^4 + 1^2 + (0^2 + 4)/2.
%e A350012 a(71) = 1 with 71 = 4*1^4 + 3^2 + (10^2 + 4^2)/2.
%e A350012 a(92) = 1 with 92 = 4*1^4 + 6^2 + (10^2 + 4)/2.
%e A350012 a(167) = 1 with 167 = 4*1^4 + 9^2 + (10^2 + 4^3)/2.
%e A350012 a(271) = 1 with 271 = 4*1^4 + 11^2 + (6^2 + 4^4)/2.
%e A350012 a(316) = 1 with 316 = 4*1^4 + 4^2 + (24^2 + 4^2)/2.
%e A350012 a(4796) = 1 with 4796 = 4*5^4 + 36^2 + (44^2 + 4^3)/2.
%e A350012 a(14716) = 1 with 14716 = 4*5^4 + 4^2 + (156^2 + 4^3)/2.
%e A350012 a(24316) = 1 with 24316 = 4*3^4 + 84^2 + (184^2 + 4^2)/2.
%t A350012 SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
%t A350012 tab={};Do[r=0;Do[If[SQ[2(n-4x^4-y^2)-4^z],r=r+1],{x,0,((n-1)/4)^(1/4)},{y,0,Sqrt[n-1-4x^4]},{z,0,Log[4,2(n-4x^4-y^2)]}];tab=Append[tab,r],{n,1,100}];Print[tab]
%Y A350012 Cf. A000118, A000290, A000302, A000583, A349661, A349957, A349992.
%K A350012 nonn
%O A350012 1,2
%A A350012 _Zhi-Wei Sun_, Dec 08 2021