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COMMENTS
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Conjecture: (i) a(n) > 0 for all n = 0,1,2,..., and a(n) = 1 only for n = 0, 1, 15, 29, 33, 47, 53, 65, 89, 129, 689, 1553, 2^(2k+1)*m (k = 0,1,2,... and m = 1, 29).
(ii) Any natural number can be written as x^2 + y^2 + z^2 + w^2 with w,x,y,z nonnegative integers and a*x^2 + b*y^2 + c*z^2 - d*w^2 a square, if (a,b,c,d) is among the following quadruples: (1,3,6,3), (1,3,9,3), (1,3,30,3), (1,4,12,4), (1,4,20,4), (1,5,20,5), (1,5,35,20), (3,4,9,3), (3,9,40,3), (4,5,16,4), (4,11,33,11), (4,12,16,7), (5,16,20,20), (5,25,36,5), (6,10,25,10), (9,12,28,12), (9,21,28,21), (15,21,25,15), (15,24,25,15), (1,5,60,5), (1,20,60,20), (9, 28,63,63), (9,28,84,84), (12,33,64,12), (16,21,105,21), (16,33,64,16), (21,25,45,45), (24,25,75,75), (24,25,96,96), (25,40,96,40), (25,48,96,48), (25,60,84,60), (25,60,96,60), (25,75,126,75), (32,64,105,32).
(iii) Any natural number can be written as x^2 + y^2 + z^2 + w^2 with w,x,y,z nonnegative integers and a*x^2 + b*y^2 - c*z^2 -d*w^2 a square, whenever (a,b,c,d) is among the quadruples (3,9,3,20), (5,9,5,20), (5,25,4,5), (9,81,9,20),(12,16,3,12), (16,64,15,16), (20,25,4,20), (27,81,20,27), (30,64,15,30), (32,64,15,32), (48,64,15,48), (60,64,15,60), (60,81,20,60), (64,80,15,80).
(iv) For each triple (a,b,c) = (21,5,15), (36,3,8), (48,8,39), (64,7,8), (40,15,144), (45,20,144), (69,20,60), any natural number can be written as x^2 + y^2 + z^2 + w^2 with w,x,y,z nonnegative integers and a*x^2 - b*y^2 - c*z^2 a square.
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