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

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, 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, 1, 60, 117, 56, 105, 48, 85, 36, 57, 20, 21, 1, 143, 35, 15, 4, 119, 3, 95, 5, 7, 11, 23

Offset: 1

Author

Stefano Spezia, Aug 04 2025

Keywords

• nonn
• easy
• frac
• look
• tabl

Examples

			The triangle of the fractions begins as:
  1/1;
  1/1,   3/5;
  1/1,   4/5,  5/13;
  1/1, 15/17,   3/5,  7/25;
  1/1, 12/13, 21/29,  8/17,  9/41;
  1/1, 35/37,   4/5,   3/5,  5/13, 11/61;
  1/1, 24/25, 45/53, 20/29, 33/65, 12/37, 13/85;
  ...
		

Links

• Stefano Spezia, First 150 rows of the triangle, flattened

Crossrefs

Cf. A000012 (k=0), A000290, A005408, A066830 (k=1), A069011, A094728, A386824 (denominators).

Programs

Mathematica

T[n_,k_]:=Numerator[(n^2-k^2)/(n^2+k^2)]; Table[T[n,k],{n,12},{k,0,n-1}]//Flatten

Formula

T(n,n-1) = A005804(n-1).