A386492 Array read by ascending antidiagonals: A(n,m) = numerator(n*(m^2 - 1)/(m^2 + 1)), where m > 0.

0, 0, 3, 0, 6, 4, 0, 9, 8, 15, 0, 12, 12, 30, 12, 0, 3, 16, 45, 24, 35, 0, 18, 4, 60, 36, 70, 24, 0, 21, 24, 75, 48, 105, 48, 63, 0, 24, 28, 90, 60, 140, 72, 126, 40, 0, 27, 32, 105, 72, 175, 96, 189, 80, 99, 0, 6, 36, 120, 84, 210, 24, 252, 120, 198, 60, 0, 33, 8, 135, 96, 245, 144, 63, 160, 297, 120, 143

Offset

1,3

Comments

A(n,m) is the numerator of 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 y-values begins as:
  0,  3/5,  4/5, 15/17, 12/13,  35/37, ...
  0,  6/5,  8/5, 30/17, 24/13,  70/37, ...
  0,  9/5, 12/5, 45/17, 36/13, 105/37, ...
  0, 12/5, 16/5, 60/17, 48/13, 140/37, ...
  0,  3/1,  4/1, 75/17, 60/13, 175/37, ...
  ...
For n = 3 and m = 2: (12/5)^2 + (9/5)^2 = 3^2.

Mathematica

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

Crossrefs

Cf. A386490 (numerator of x), A386491 (denominator of x and y).

Cf. A002522.

Sequence in context: A088162 A133170 A062542 * A360173 A109693 A381756

Adjacent sequences: A386489 A386490 A386491 * A386493 A386494 A386495

Keyword

nonn,easy,frac,tabl

Author

Stefano Spezia, Jul 23 2025

Status

approved