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A088307 Triangle, read by rows, T(n,k) = n^2 + k^2 if gcd(n,k)=1, otherwise 0.

%I A088307 #18 Feb 16 2025 08:32:51
%S A088307 2,5,0,10,13,0,17,0,25,0,26,29,34,41,0,37,0,0,0,61,0,50,53,58,65,74,
%T A088307 85,0,65,0,73,0,89,0,113,0,82,85,0,97,106,0,130,145,0,101,0,109,0,0,0,
%U A088307 149,0,181,0,122,125,130,137,146,157,170,185,202,221,0
%N A088307 Triangle, read by rows, T(n,k) = n^2 + k^2 if gcd(n,k)=1, otherwise 0.
%C A088307 (n^2-k^2, 2*k*n, T(n,k)) is a primitive Pythagorean triple iff T(n,k) > 0.
%H A088307 G. C. Greubel, <a href="/A088307/b088307.txt">Rows n = 1..50 of the triangle, flattened</a>
%H A088307 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PythagoreanTriple.html">Pythagorean Triple</a>
%F A088307 T(n, n) = 2*A000007(n-1).
%F A088307 T(n, 1) = A002522(n).
%F A088307 T(2*n+1, 2) = A078370(n).
%F A088307 Sum_{k=1..n} A057427(T(n,m)) = A000010(n).
%F A088307 From _G. C. Greubel_, Dec 15 2022: (Start)
%F A088307 T(n, n-1) = A001844(n).
%F A088307 T(n, n-2) = ((1-(-1)^n)/2) * A008527((n+1)/2).
%F A088307 T(2*n, n) = 5*A000007(n-1).
%F A088307 T(2*n+1, n) = A079273(n+1).
%F A088307 T(2*n-1, n) = A190816(n).
%F A088307 Sum_{k=1..floor((n+1)/2)} T(n-k+1, k) = A053818(n+1) + [n=1]. (End)
%e A088307 Triangle begins:
%e A088307    2;
%e A088307    5,  0;
%e A088307   10, 13,  0;
%e A088307   17,  0, 25,  0;
%e A088307   26, 29, 34, 41,  0;
%e A088307   37,  0,  0,  0, 61, 0;
%e A088307   ...
%t A088307 Table[If[CoprimeQ[n,k],n^2+k^2,0],{n,20},{k,n}]//Flatten (* _Harvey P. Dale_, Jul 13 2018 *)
%o A088307 (Magma)
%o A088307 function A088307(n,k)
%o A088307   if GCD(k,n) eq 1 then return n^2+k^2;
%o A088307   else return 0;
%o A088307   end if; return A088307;
%o A088307 end function;
%o A088307 [A088307(n,k): k in [1..n], n in [1..13]]; // _G. C. Greubel_, Dec 16 2022
%o A088307 (SageMath)
%o A088307 def A088307(n,k):
%o A088307     if (gcd(n,k)==1): return n^2 + k^2
%o A088307     else: return 0
%o A088307 flatten([[A088307(n,k) for k in range(1,n+1)] for n in range(1,14)]) # _G. C. Greubel_, Dec 16 2022
%Y A088307 Cf. A000007, A000010, A001844, A002522, A008527, A053818.
%Y A088307 Cf. A057427, A070216, A078370, A079273, A190816.
%K A088307 nonn,tabl
%O A088307 1,1
%A A088307 _Reinhard Zumkeller_, Nov 05 2003