A386491 Array read by ascending antidiagonals: A(n,m) = denominator(2n*m/(m^2 + 1)), where m > 0.

1, 1, 5, 1, 5, 5, 1, 5, 5, 17, 1, 5, 5, 17, 13, 1, 1, 5, 17, 13, 37, 1, 5, 1, 17, 13, 37, 25, 1, 5, 5, 17, 13, 37, 25, 65, 1, 5, 5, 17, 13, 37, 25, 65, 41, 1, 5, 5, 17, 13, 37, 25, 65, 41, 101, 1, 1, 5, 17, 13, 37, 5, 65, 41, 101, 61, 1, 5, 1, 17, 13, 37, 25, 13, 41, 101, 61, 145

Offset

1,3

Comments

A(n,m) is the denominator of x and y in the m-th rational solution (x,y) to the equation x^2 + y^2 = n^2 given by Diophantus in Book II of Arithmetica.

References

James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 239, 253.

Links

Table of n, a(n) for n=1..78.

Example

The array of the rational x-values begins as:
  1/1,  4/5,  3/5,  8/17,  5/13, 12/37, ...
  2/1,  8/5,  6/5, 16/17, 10/13, 24/37, ...
  3/1, 12/5,  9/5, 24/17, 15/13, 36/37, ...
  4/1, 16/5, 12/5, 32/17, 20/13, 48/37, ...
  5/1,  4/1,  3/1, 40/17, 25/13, 60/37, ...
  ...
For n = 3 and m = 2: (12/5)^2 + (9/5)^2 = 3^2.

Mathematica

A[n_, m_]:=Denominator[2n*m/(m^2+1)]; Table[A[n-m+1, m], {n, 12}, {m, n}]//Flatten

Crossrefs

Cf. A386490 (numerator of x), A386492 (numerator of x).

Cf. A002522, A003991.

Sequence in context: A021198 A388739 A275976 * A306577 A143969 A198366

Adjacent sequences: A386488 A386489 A386490 * A386492 A386493 A386494

Keyword

nonn,easy,frac,tabl

Author

Stefano Spezia, Jul 23 2025

Status

approved