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%I #26 Apr 18 2018 04:48:45
%S 0,0,0,1,2,1,2,4,3,3,5,4,6,7,4,7,5,4,7,8,5,5,8,5,9,7,6,13,10,7,9,10,7,
%T 12,11,8,11,7,7,11,11,6,11,13,6,10,7,7,17,13,6,13,14,9,11,18,10,13,14,
%U 11
%N Number of ways to write n as x^2 + 5*y^2 + 2^z + 3*2^w with x,y,z,w nonnegative integers.
%C Conjecture: a(n) > 0 for all n > 3.
%C Clearly, a(4*n) > 0 if a(n) > 0. We have verified a(n) > 0 for all n = 4..2*10^8.
%C See also A302983 and A302984 for similar conjectures.
%H Zhi-Wei Sun, <a href="/A302982/b302982.txt">Table of n, a(n) for n = 1..10000</a>
%H 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 Zhi-Wei Sun, <a href="http://maths.nju.edu.cn/~zwsun/179b.pdf">New conjectures on representations of integers (I)</a>, Nanjing Univ. J. Math. Biquarterly 34(2017), no. 2, 97-120.
%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1701.05868">Restricted sums of four squares</a>, arXiv:1701.05868 [math.NT], 2017-2018.
%e a(4) = 1 with 4 = 0^2 + 5*0^2 + 2^0 + 3*2^0.
%e a(5) = 2 with 5 = 1^2 + 5*0^2 + 2^0 + 3*2^0 = 0^2 + 5*0^2 + 2^1 + 3*2^0.
%e a(6) = 1 with 6 = 1^2 + 3*0^2 + 2^1 + 3*2^0.
%t SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
%t tab={};Do[r=0;Do[If[SQ[n-3*2^k-2^j-5x^2],r=r+1],{k,0,Log[2,n/3]},{j,0,If[n==3*2^k,-1,Log[2,n-3*2^k]]},{x,0,Sqrt[(n-3*2^k-2^j)/5]}];tab=Append[tab,r],{n,1,60}];Print[tab]
%Y Cf. A000079, A000290, A020669, A271518, A281976, A299924, A299537, A299794, A300219, A300362, A300396, A300441, A301376, A301391, A301471, A301472, A301534, A302920, A302981, A302983, A302984.
%K nonn
%O 1,5
%A _Zhi-Wei Sun_, Apr 16 2018
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