cp's OEIS Frontend

A273110 Number of ordered ways to write n as x^2 + y^2 + z^2 + w^2 with (x+4*y+4*z)^2 + (9*x+3*y+3*z)^2 a square, where x,y,z,w are nonnegative integers with y > 0 and y >= z <= w.

%I A273110 #21 May 18 2016 23:24:59
%S A273110 1,2,2,1,2,3,1,2,3,3,3,2,2,2,2,1,5,6,2,2,2,3,1,3,3,4,6,1,4,4,1,2,6,5,
%T A273110 3,3,2,5,1,3,6,5,4,3,4,3,1,2,4,7,7,2,4,8,1,2,6,3,4,2,4,5,4,1,7,8,4,5,
%U A273110 4,4,1,6,5,7,5,2,4,5,1,2
%N A273110 Number of ordered ways to write n as x^2 + y^2 + z^2 + w^2 with (x+4*y+4*z)^2 + (9*x+3*y+3*z)^2 a square, where x,y,z,w are nonnegative integers with y > 0 and y >= z <= w.
%C A273110 Conjecture: (i) a(n) > 0 for all n > 0, and a(n) = 1 only for n = 4^k*m (k = 0,1,2,... and m = 1, 7, 23, 31, 39, 47, 55, 71, 79, 119, 151, 191, 311, 671).
%C A273110 (ii) Any natural number can be written as x^2 + y^2 + z^2 + w^2 with (x+y+z)^2 + (4*(x+y-z))^2 a square, where x,y,z,w are nonnegative integers with x+y >= z.
%C A273110 (iii) For each tuple (a,b,c,d,e,f) = (1,1,1,3,6,-3), (1,1,1,4,12,-12), (1,1,2,1,1,-5), (1,1,2,1,8,-5), (1,1,2,3,3,-3), (1,1,2,4,4,-8), (1,3,11,12,4,4), (1,3,14,16,4,4), (1,3,14,18,4,2), (1,3,20,16,4,12), (1,4,11,6,3,3), (1,5,13,12,12,12), (1,5,14,15,12,21), (1,6,6,16,8,8), (1,6,14,12,8,8), (1,6,14,16,8,4), (1,6,17,20,8,4), (1,6,20,20,8,8), (1,7,8,4,2,6), (1,7,8,10,5,15), (1,7,9,10,5,12), (1,7,15,4,2,8), (1,7,15,10,5,20), any natural number can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers such that (a*x+b*y+c*z)^2 + (d*x+e*y+f*z)^2 is a square.
%C A273110 It was proved in arXiv:1604.06723 that any positive integer can be written as x^2 + y^2 + z^2 + w^2 with x,y,z,w nonnegative integers and y > 0 such that x+4*y+4*z and 9*x+3*y+3*z are the two legs of a right triangle with positive integer sides.
%C A273110 See also A271714, A273107, A273108 and A273134 for similar conjectures related to Pythagorean triples. For more conjectural refinements of Lagrange's four-square theorem, one may consult arXiv:1604.06723.
%H A273110 Zhi-Wei Sun, <a href="/A273110/b273110.txt">Table of n, a(n) for n = 1..10000</a>
%H A273110 Zhi-Wei Sun, <a href="http://arxiv.org/abs/1604.06723">Refining Lagrange's four-square theorem</a>, arXiv:1604.06723 [math.GM], 2016.
%e A273110 a(1) = 1 since 1 = 0^2 + 1^2 + 0^2 + 0^2 with 1 > 0 = 0 and (0+4*1+4*0)^2 + (9*0+3*1+3*0)^2 = 5^2.
%e A273110 a(7) = 1 since 7 = 2^2 + 1^2 + 1^2 + 1^2 with 0 < 1 = 1 = 1 and (2+4*1+4*1)^2 + (9*2+3*1+3*1)^2 = 26^2.
%e A273110 a(23) = 1 since 23 = 3^2 + 2^2 + 1^2 + 3^2 with 2 > 1 < 3 and (3+4*2+4*1)^2 + (9*3+3*2+3*1)^2 = 39^2.
%e A273110 a(31) = 1 since 31 = 2^2 + 1^2 + 1^2 + 5^2 with 0 < 1 = 1 < 5 and (2+4*1+4*1)^2 + (9*2+3*1+3*1)^2 = 26^2.
%e A273110 a(39) = 1 since 39 = 3^2 + 2^2 + 1^2 + 5^2 with 2 > 1 < 5 and (3+4*2+4*1)^2 + (9*3+3*2+3*1)^2 = 39^2.
%e A273110 a(47) = 1 since 47 = 5^2 + 3^2 + 2^2 + 3^2 with 3 > 2 < 3 and (5+4*3+4*2)^2 + (9*5+3*3+3*2)^2 = 65^2.
%e A273110 a(55) = 1 since 55 = 2^2 + 1^2 + 1^2 + 7^2 with 0 < 1 = 1 < 7 and (2+4*1+4*1)^2 + (9*2+3*1+3*1)^2 = 26^2.
%e A273110 a(71) = 1 since 71 = 6^2 + 5^2 + 1^2 + 3^2 with 5 > 1 < 3 and (6+4*5+4*1)^2 + (9*6+3*5+3*1)^2 = 78^2.
%e A273110 a(79) = 1 since 79 = 6^2 + 3^2 + 3^2 + 5^2 with 0 < 3 = 3 < 5 and (6+4*3+4*3)^2 + (9*6+3*3+3*3)^2 = 78^2.
%e A273110 a(119) = 1 since 119 = 5^2 + 3^2 + 2^2 + 9^2 with 3 > 2 < 9 and (5+4*3+4*2)^2 + (9*5+3*3+3*2)^2 = 65^2.
%e A273110 a(151) = 1 since 151 = 9^2 + 6^2 + 3^2 + 5^2 with 6 > 3 < 5 and (9+4*6+4*3)^2 + (9*9+3*6+3*3)^2 = 117^2.
%e A273110 a(191) = 1 since 191 = 10^2 + 9^2 + 1^2 + 3^2 with 9 > 1 < 3 and (10+4*9+4*1)^2 + (9*10+3*9+3*1)^2 = 130^2.
%e A273110 a(311) = 1 since 311 = 7^2 + 6^2 + 1^2 + 15^2 with 6 > 1 < 15 and (7+4*6+4*1)^2 + (9*7+3*6+3*1)^2 = 91^2.
%e A273110 a(671) = 1 since 671 = 17^2 + 11^2 + 6^2 + 15^2 with 11 > 6 < 15 and (17+4*11+4*6)^2 + (9*17+3*11+3*6)^2 = 221^2.
%t A273110 SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
%t A273110 Do[r=0;Do[If[SQ[n-x^2-y^2-z^2]&&SQ[(x+4y+4z)^2+(9x+3y+3z)^2],r=r+1],{x,0,Sqrt[n]},{z,0,Sqrt[(n-x^2)/3]},{y,Max[1,z],Sqrt[n-x^2-2z^2]}];Print[n," ",r];Continue,{n,1,80}]
%Y A273110 Cf. A000118, A000290, A260625, A261876, A262357, A267121, A268197, A268507, A269400, A270073, A271510, A271513, A271518, A271608, A271665, A271714, A271721, A271724, A271775, A271778, A271824, A272084, A272332, A272351, A272620, A272888, A272977, A273021, A273107, A273108, A273134.
%K A273110 nonn
%O A273110 1,2
%A A273110 _Zhi-Wei Sun_, May 15 2016