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A386823 Triangle read by rows: T(n,k) = numerator((n^2 - k^2)/(n^2 + k^2)), where 0 <= k < n.

%I A386823 #13 Aug 07 2025 00:35:56
%S A386823 1,1,3,1,4,5,1,15,3,7,1,12,21,8,9,1,35,4,3,5,11,1,24,45,20,33,12,13,1,
%T A386823 63,15,55,3,39,7,15,1,40,77,4,65,28,5,16,17,1,99,12,91,21,3,8,51,9,19,
%U A386823 1,60,117,56,105,48,85,36,57,20,21,1,143,35,15,4,119,3,95,5,7,11,23
%N A386823 Triangle read by rows: T(n,k) = numerator((n^2 - k^2)/(n^2 + k^2)), where 0 <= k < n.
%H A386823 Stefano Spezia, <a href="/A386823/b386823.txt">First 150 rows of the triangle, flattened</a>
%F A386823 T(n,n-1) = A005804(n-1).
%e A386823 The triangle of the fractions begins as:
%e A386823   1/1;
%e A386823   1/1,   3/5;
%e A386823   1/1,   4/5,  5/13;
%e A386823   1/1, 15/17,   3/5,  7/25;
%e A386823   1/1, 12/13, 21/29,  8/17,  9/41;
%e A386823   1/1, 35/37,   4/5,   3/5,  5/13, 11/61;
%e A386823   1/1, 24/25, 45/53, 20/29, 33/65, 12/37, 13/85;
%e A386823   ...
%t A386823 T[n_,k_]:=Numerator[(n^2-k^2)/(n^2+k^2)]; Table[T[n,k],{n,12},{k,0,n-1}]//Flatten
%Y A386823 Cf. A000012 (k=0), A000290, A005408, A066830 (k=1), A069011, A094728, A386824 (denominators).
%K A386823 nonn,easy,frac,look,tabl
%O A386823 1,3
%A A386823 _Stefano Spezia_, Aug 04 2025