cp's OEIS Frontend

A271644 Number of ordered ways to write n as w^2 + x^2 + y^2 + z^2 such that w*x + 2*x*y + 2*y*z is a square, where w is a positive integer and x,y,z are nonnegative integers.

%I A271644 #21 Mar 11 2023 08:05:19
%S A271644 1,3,1,1,4,4,1,3,4,4,2,2,5,2,1,1,8,8,2,5,7,3,2,4,8,7,3,2,6,4,4,3,7,6,
%T A271644 2,4,6,4,3,4,9,4,3,4,8,4,1,2,5,7,4,7,10,11,3,2,5,5,2,2,7,4,2,1,8,9,2,
%U A271644 8,14,9,1,8,8,6,5,4,8,2,3,5
%N A271644 Number of ordered ways to write n as w^2 + x^2 + y^2 + z^2 such that w*x + 2*x*y + 2*y*z is a square, where w is a positive integer and x,y,z are nonnegative integers.
%C A271644 Conjecture: (i) a(n) > 0 for all n > 0, and a(n) = 1 only for n = 3, 7, 15, 47, 71, 379, 4^k (k = 0,1,2,...).
%C A271644 (ii) If a, b and c are positive integers with a <= b <= c, gcd(a,b,c) squarefree, and the triple (a,b,c) not equal to (1,2,2), then not all natural numbers can be written as w^2 + x^2 + y^2 + z^2 with w,x,y,z nonnegative integers and a*w*x + b*x*y + c*y*z a square.
%C A271644 (iii) Let a,b,c be positive integers with gcd(a,b,c) squarefree. Then every n = 0,1,2,... can be written as w^2 + x^2 + y^2 + z^2 with w,x,y,z nonnegative integers such that a*x*y + b*y*z + c*z*x is a square, if and only if {a,b,c} is among {1,2,3}, {1,3,8}, {1,8,13}, {2,4,45}, {4,5,7}, {4,7,23}, {5,8,9}, {11,16,31}.
%C A271644 Clearly, part (i) of this conjecture is stronger than Lagrange's four-square theorem.
%C A271644 See also A271510, A271513, A271518 and A271608 for related conjectures.
%H A271644 Zhi-Wei Sun, <a href="/A271644/b271644.txt">Table of n, a(n) for n = 1..10000</a>
%H A271644 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 A271644 a(1) = 1 since 1 = 1^2 + 0^2 + 0^2 + 0^2 with 1*0 + 2*0*0 + 2*0*0 = 0^2.
%e A271644 a(3) = 1 since 3 = 1^2 + 1^2 + 0^2 + 1^2 with 1*1 + 2*1*0 + 2*0*1 = 1^2.
%e A271644 a(7) = 1 since 7 = 1^2 + 1^2 + 2^2 + 1^2 with 1*1 + 2*1*2 + 2*2*1 = 3^2.
%e A271644 a(11) = 2 since 11 = 1^2 + 1^2 + 0^2 + 3^2 with 1*1 + 2*1*0 + 2*0*3 = 1^2, and 11 = 1^2 + 3^2 + 1^2 + 0^2 with 1*3 + 2*3*1 + 2*1*0 = 3^2.
%e A271644 a(12) = 2 since 12 = 1^2 + 1^2 + 1^2 + 3^2 with 1*1 + 2*1*1 + 2*1*3 = 3^2, and 12 = 2^2 + 2^2 + 0^2 + 2^2  with 2*2 + 2*2*0 + 2*0*2 = 2^2.
%e A271644 a(15) = 1 since 15 = 3^2 + 1^2 + 1^2 + 2^2 with 3*1 + 2*1*1 + 2*1*2 = 3^2.
%e A271644 a(47) = 1 since 47 = 1^2 + 1^2 + 6^2 + 3^2 with 1*1 + 2*1*6 + 2*6*3 = 7^2.
%e A271644 a(71) = 1 since 71 = 3^2 + 3^2 + 2^2 + 7^2 with 3*3 + 2*3*2 + 2*2*7 = 7^2.
%e A271644 a(379) = 1 since 379 = 3^2 + 3^2 + 0^2 + 19^2 with 3*3 + 2*3*0 + 2*0*19 = 3^2.
%t A271644 SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
%t A271644 Do[r=0;Do[If[SQ[n-w^2-x^2-y^2]&&SQ[w*x+2*x*y+2*y*Sqrt[n-w^2-x^2-y^2]],r=r+1],{w,1,Sqrt[n]},{x,0,Sqrt[n-w^2]},{y,0,Sqrt[n-w^2-x^2]}];Print[n," ",r];Continue,{n,1,80}]
%Y A271644 Cf. A000118, A000290, A271510, A271513, A271518, A271608.
%K A271644 nonn
%O A271644 1,2
%A A271644 _Zhi-Wei Sun_, Apr 11 2016