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A343897 Number of ways to write n as 2*x + y + z with x,y,z positive integers such that 16*x^2*y^2 + 19*y^2*z^2 + 29*z^2*x^2 is a square.

%I A343897 #12 May 07 2021 22:49:51
%S A343897 0,0,0,1,1,1,1,1,1,2,3,2,1,2,4,2,1,3,1,3,1,5,2,2,6,1,1,5,5,6,2,2,4,2,
%T A343897 4,5,6,3,2,3,2,5,2,7,10,4,1,3,3,10,9,2,5,5,10,6,7,6,7,8,8,4,7,4,5,8,2,
%U A343897 4,4,13,9,5,6,10,11,6,11,6,6,5,4,4,10,9,8,8,8,8,9,16,5,5,6,4,7,3,12,7,11,13
%N A343897 Number of ways to write n as 2*x + y + z with x,y,z positive integers such that 16*x^2*y^2 + 19*y^2*z^2 + 29*z^2*x^2 is a square.
%C A343897 Conjecture: a(n) > 0 for all n > 3.
%C A343897 We have verified a(n) > 0 for all n = 4..10000. The conjecture holds if a(p) > 0 for each odd prime p.
%C A343897 It seems that a(n) = 1 only for n = 4..9, 13, 17, 19, 21, 26, 27, 47.
%H A343897 Zhi-Wei Sun, <a href="/A343897/b343897.txt">Table of n, a(n) for n = 1..1500</a>
%e A343897 a(4) = 1 with 4 = 2*1 + 1 + 1 and 16*1^2*1^2 + 19*1^2*1^2 + 29*1^2*1^2 = 8^2.
%e A343897 a(6) = 1 with 6 = 2*1 + 2 + 2 and 16*1^2*2^2 + 19*2^2*2^2 + 29*2^2*1^2 = 22^2.
%e A343897 a(9) = 1 with 9 = 2*2 + 4 + 1 and 16*2^2*4^2 + 19*4^2*1^2 + 29*1^2*2^2 = 38^2.
%e A343897 a(13) = 1 with 13 = 2*5 + 2 + 1 and 16*5^2*2^2 + 19*2^2*1^2 + 29*1^2*5^2 = 49^2.
%e A343897 a(19) = 1 with 19 = 2*2 + 14 + 1 and 16*2^2*14^2 + 19*14^2*1^2 + 29*1^2*2^2 = 128^2.
%e A343897 a(47) = 1 with 47 = 2*13 + 13 + 8 and 16*13^2*13^2 + 19*13^2*8^2 + 29*8^2*13^2 = 988^2.
%t A343897 SQ[n_]:=IntegerQ[Sqrt[n]];
%t A343897 tab={};Do[r=0;Do[If[SQ[16(x*y)^2+(n-2x-y)^2*(19*y^2+29x^2)],r=r+1],{x,1,(n-2)/2},{y,1,n-1-2x}];tab=Append[tab,r],{n,1,100}];Print[tab]
%Y A343897 Cf. A000041, A000290, A230121, A230747, A340274, A343862.
%K A343897 nonn
%O A343897 1,10
%A A343897 _Zhi-Wei Sun_, May 03 2021