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%I #21 May 01 2016 13:24:11
%S 1,1,1,1,2,2,2,1,3,3,2,3,4,3,1,1,4,5,2,3,3,4,4,2,5,5,2,5,5,2,1,1,3,6,
%T 2,3,4,8,1,3,8,7,3,3,4,5,2,3,6,9,4,6,10,4,3,3,3,8,5,4,5,5,5,1,7,4,2,7,
%U 4,5,1,5,7,5,2,4,8,1,1,3
%N Number of ordered ways to write n as w^2 + x^2 + y^2 + z^2 with 6*w^2*x^2 + 12*x^2*y^2 + 52*y^2*z^2 + 27*z^2*w^2 a square, where w,x,y are nonnegative integers and z is a positive integer.
%C Conjecture: (i) a(n) > 0 for all n > 0, and a(n) = 1 only for n = 3, 15, 31, 39, 71, 79, 195, 311, 319, 403, 559, 591, 683, 719, 1031, 1439, 1643, 2519, 6879, 2^k, 2^(2k+1)*39 (k = 0,1,2,...). Also, any positive integer can be written as w^2 + x^2 + y^2 + z^2 with x a positive integer and w,y,z nonnegative integer such that 6*w^2*x^2 + 12*x^2*y^2 + 52*y^2*z^2 + 27*z^2*w^2 is a square.
%C (ii) For each triple (a,b,c) = (1,3,2), (1,11,9), (1,14,4),(1,20,25), (1,27,18), (1,36,9), (1,56,4), (4,32,25), (9,15,25), (9,35,25), (25,8,64), (25,15,54), (25,32,28), (25,35,49), (28,32,49), any natural number can be written as w^2 + x^2 + y^2 + z^2 with w,x,y,z integers such that a*w^2*x^2 + b*x^2*y^2 + c*y^2*z^2 is a square.
%C See also A268507, A271510, A271513, A271518, A271665, A271714, A271721, A271724, A271775, A271778 and A271824 for other conjectures refining Lagrange's four-square theorem.
%H Zhi-Wei Sun, <a href="/A269400/b269400.txt">Table of n, a(n) for n = 1..10000</a>
%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1604.06723">Refining Lagrange's four-square theorem</a>, arXiv:1604.06723, 2016.
%e a(1) = 1 since 1 = 0^2 + 0^2 + 0^2 + 1^2 with 1 > 0 and 6*0^2*0^2 + 12*0^2*0^2 + 52*0^2*1^2 + 27*1^2*0^2 = 0^2.
%e a(2) = 1 since 2 = 0^2 + 1^2 + 0^2 + 1^2 with 1 > 0 and 6*0^2*1^2 + 12*1^2*0^2 + 52*0^2*1^2 + 27*1^2*0^2 = 0^2.
%e a(3) = 1 since 3 = 0^2 + 1^2 + 1^2 + 1^2 with 1 > 0 and 6*0^2*1^2 + 12*1^2*1^2 + 52*1^2*1^2 + 27*1^2*0^2 = 8^2.
%e a(15) = 1 since 15 = 2^2 + 3^2 + 1^2 + 1^2 with 1 > 0 and 6*2^2*3^2 + 12*3^2*1^2 + 52*1^2*1^2 = 22^2.
%e a(31) = 1 since 31 = 1^2 + 1^2 + 2^2 + 5^2 with 5 > 0 and 6*1^2*1^2 + 12*1^2*2^2 + 52*2^2*5^2 + 27*5^2*1^2 = 77^2.
%e a(39) = 1 since 39 = 2^2 + 1^2 + 5^2 + 3^2 with 3 > 0 and 6*2^2*1^2 + 12*1^2*5^2 + 52*5^2*3^2 + 27*3^2*2^2 = 114^2.
%e a(71) = 1 since 71 = 3^2 + 1^2 + 6^2 + 5^2 with 5 > 0 and 6*3^2*1^2 + 12*1^2*6^2 + 52*6^2*5^2 + 27*5^2*3^2 = 231^2.
%e a(78) = 1 since 78 = 2^2 + 7^2 + 4^2 + 3^2 with 3 > 0 and 6*2^2*7^2 + 12*7^2*4^2 + 52*7^2*4^2 + 27*3^2*2^2 = 138^2.
%e a(79) = 1 since 79 = 2^2 + 5^2 + 7^2 + 1^2 with 1 > 0 and 6*2^2*5^2 + 12*5^2*7^2 + 52*7^2*1^2 + 27*1^2*2^2 = 134^2.
%e a(195) = 1 since 195 = 3^2 + 7^2 + 4^2 + 11^2 with 11 > 0 and 6*3^2*7^2 + 12*7^2*4^2 + 52*4^2*11^2 + 27*11^2*3^2 = 377^2.
%e a(311) = 1 since 311 = 14^2 + 9^2 + 3^2 + 5^2 with 5 > 0 and 6*14^2*9^2 + 12*9^2*3^2 + 52*3^2*5^2 + 27*5^2*14^2 = 498^2.
%e a(319) = 1 since 319 = 6^2 + 3^2 + 7^2 + 15^2 with 15 > 0 and 6*6^2*3^2 + 12*3^2*7^2 + 52*7^2*15^2 + 27*15^2*6^2 = 894^2.
%e a(403) = 1 since 403 = 3^2 + 13^2 + 12^2 + 9^2 with 9 > 0 and 6*3^2*13^2 + 12*13^2*12^2 + 52*12^2*9^2 + 27*9^2*3^2 = 963^2.
%e a(559) = 1 since 559 = 5^2 + 23^2 + 2^2 + 1^2 with 1 > 0 and 6*5^2*23^2 + 12*23^2*2^2 + 52*2^2*1^2 + 27*1^2*5^2 = 325^2.
%e a(591) = 1 since 591 = 21^2 + 11^2 + 2^2 + 5^2 with 5 > 0 and 6*21^2*11^2 + 12*11^2*2^2 + 52*2^2*5^2 + 27*5^2*21^2 = 793^2.
%e a(683) = 1 since 683 = 0^2 + 11^2 + 21^2 + 11^2 with 11 > 0 and 6*0^2*11^2 + 12*11^2*21^2 + 52*21^2*11^2 + 27*11^2*0^2 = 1848^2.
%e a(719) = 1 since 719 = 10^2 + 3^2 + 21^2 + 13^2 with 13 > 0 and 6*10^2*3^2 + 12*3^2*21^2 + 52*21^2*13^2 + 27*13^2*10^2 = 2094^2.
%e a(1031) = 1 since 1031 = 26^2 + 15^2 + 9^2 + 7^2 with 7 > 0 and 6*26^2*15^2 + 12*15^2*9^2 + 52*9^2*7^2 + 27*7^2*26^2 = 1494^2.
%e a(1439) = 1 since 1439 = 13^2 + 27^2 + 10^2 + 21^2 with 21 > 0 and 6*13^2*27^2 + 12*27^2*10^2 + 52*10^2*21^2 + 27*21^2*13^2 = 2433^2.
%e a(1643) = 1 since 1643 = 36^2 + 17^2 + 3^2 + 7^2 with 7 > 0 and 6*36^2*17^2 + 12*17^2*3^2 + 52*3^2*7^2 + 27*7^2*36^2 = 2004^2.
%e a(2519) = 1 since 2519 = 27^2 + 7^2 + 30^2 + 29^2 with 29 > 0 and 6*27^2*7^2 + 12*7^2*30^2 + 52*30^2*29^2 + 27*29^2*27^2 = 7527^2.
%e a(6879) = 1 since 6879 = 38^2 + 53^2 + 49^2 + 15^2 with 15 > 0 and 6*38^2*53^2 + 12*53^2*49^2 + 52*49^2*15^2 + 27*15^2*38^2 = 11922^2.
%t SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
%t Do[r=0;Do[If[SQ[n-x^2-y^2-z^2]&&SQ[6*(n-x^2-y^2-z^2)*x^2+12*x^2*y^2+52*y^2*z^2+27*z^2*(n-x^2-y^2-z^2)],r=r+1],{x,0,Sqrt[n-1]},{y,0,Sqrt[n-1-x^2]},{z,1,Sqrt[n-x^2-y^2]}];Print[n," ",r];Continue,{n,1,80}]
%Y Cf. A000118, A000290, A268507, A271510, A271513, A271518, A271608, A271665, A271714, A271721, A271724, A271775, A271778, A271824.
%K nonn
%O 1,5
%A _Zhi-Wei Sun_, Apr 16 2016
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