cp's OEIS Frontend

A271714 Number of ordered ways to write n as w^2 + x^2 + y^2 + z^2 such that (10*w+5*x)^2 + (12*y+36*z)^2 is a square, where w is a positive integer and x,y,z are nonnegative integers.

%I A271714 #27 May 01 2016 13:27:24
%S A271714 1,1,1,1,3,1,1,1,1,3,2,1,3,1,2,1,2,3,1,4,4,2,2,1,3,3,5,2,2,5,2,1,2,3,
%T A271714 3,3,2,3,2,3,4,4,2,3,9,2,3,1,1,6,2,3,4,6,4,1,2,5,3,3,4,3,5,1,4,5,1,3,
%U A271714 6,6,1,3,4,5,12,2,4,6,2,4
%N A271714 Number of ordered ways to write n as w^2 + x^2 + y^2 + z^2 such that (10*w+5*x)^2 + (12*y+36*z)^2 is a square, where w is a positive integer and x,y,z are nonnegative integers.
%C A271714 Conjecture: (i) a(n) > 0 for all n > 0, and a(n) = 1 only for n = 7, 9, 19, 49, 133, 589, 2^k, 2^k*3, 4^k*q (k = 0,1,2,... and q = 14, 67, 71, 199).
%C A271714 (ii) If P(y,z) is one of 2y-3z, 2y-8z and 4y-6z, then any natural number can be written as w^2 + x^2 + y^2 + z^2 with w,x,y,z nonnegative integers such that (w-x)^2 + P(y,z)^2 is a square.
%C A271714 (iii) For each triple (a,b,c) = (1,4,4), (1,12,12), (2,4,8), (2,6,6), (2,12,12), (3,4,4), (3,4,8), (3,8,8), (3,12,12), (3,12,36), (5,4,4), (5,4,8), (5,8,16), (5,36,36), (6,4,4), (7,12,12), (7,20,20), (7,24,24), (9,4,4), (9,12,12),(9,36,36), (11,12,12), (13,4,4), (15,12,12), (16,12,12), (21,20,20), (21,24,24), (23,12,12), any natural number can be written as w^2 + x^2 + y^2 + z^2 with w,x,y,z nonnegative integers such that (w+a*x)^2 + (b*y-c*z)^2 is a square.
%C A271714 See also A271510, A271513, A271518, A271644, A271665, A271721 and A271724 for other conjectures refining Lagrange's four-square theorem.
%H A271714 Zhi-Wei Sun, <a href="/A271714/b271714.txt">Table of n, a(n) for n = 1..10000</a>
%H A271714 Zhi-Wei Sun, <a href="http://arxiv.org/abs/1604.06723">Refining Lagrange's four-square theorem</a>, arXiv:1604.06723, 2016.
%e A271714 a(2) = 1 since 2 = 1^2 + 1^2 + 0^2 + 0^2 with (10*1+5*1)^2 + (12*0+36*0)^2 = 15^2 + 0^2 = 15^2.
%e A271714 a(3) = 1 since 3 = 1^2 + 1^2 + 0^2 + 1^2 with (10*1+5*1)^2 + (12*0+36*1)^2 = 15^2 + 36^2 = 39^2.
%e A271714 a(4) = 1 since 4 = 2^2 + 0^2 + 0^2 + 0^2 with (10*2+5*0)^2 + (12*0+36*0)^2 = 20^2 + 0^2 = 20^2.
%e A271714 a(6) = 1 since 6 = 2^2 + 0^2 + 1^2 + 1^2 with (10*2+5*0)^2 + (12*1+36*1)^2 = 20^2 + 48^2 = 52^2.
%e A271714 a(7) = 1 since 7 = 1^2 + 2^2 + 1^2 + 1^2 with (10*1+5*2)^2 + (12*1+36*1)^2 = 20^2 + 48^2 = 52^2.
%e A271714 a(9) = 1 since 9 = 3^2 + 0^2 + 0^2 + 0^2 with (10*3+5*0)^2 + (12*0+36*0)^2 = 30^2 + 0^2 = 30^2.
%e A271714 a(19) = 1 since 19 = 3^2 + 0^2 + 3^2 + 1^2 with (10*3+5*0)^2 + (12*3+36*1)^2 = 30^2 + 72^2 = 78^2.
%e A271714 a(49) = 1 since 49 = 7^2 + 0^2 + 0^2 + 0^2 with (10*7+5*0)^2 + (12*0+36*0)^2 = 70^2 + 0^2 = 70^2.
%e A271714 a(133) = 1 since 133 = 9^2 + 0^2 + 6^2 + 4^2 with (10*9+5*0)^2 + (12*6+36*4)^2 = 90^2 + 216^2 = 234^2.
%e A271714 a(589) = 1 since 589 = 17^2 + 10^2 + 2^2 + 14^2 with (10*17+5*10)^2 + (12*2+36*14)^2 = 220^2 + 528^2 = 572^2.
%t A271714 SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]]
%t A271714 Do[r=0;Do[If[SQ[n-x^2-y^2-z^2]&&SQ[(10*Sqrt[n-x^2-y^2-z^2]+5x)^2+(12y+36z)^2],r=r+1],{x,0,Sqrt[n-1]},{y,0,Sqrt[n-1-x^2]},{z,0,Sqrt[n-1-x^2-y^2]}];Print[n," ",r];Continue,{n,1,80}]
%Y A271714 Cf. A000118, A000290, A271510, A271513, A271518, A271608, A271644, A271665, A271719, A271721, A271724.
%K A271714 nonn
%O A271714 1,5
%A A271714 _Zhi-Wei Sun_, Apr 12 2016