A384440 Array of triples (x,y,z) of minimal (positive) solutions of the cubic Pell equation x^3 + n*y^3 + n^2*z^3 - 3*n*x*y*z = 1, read by rows.

1, 0, 0, 1, 1, 1, 4, 3, 2, 5, 3, 2, 41, 24, 14, 109, 60, 33, 4, 2, 1, 1, 0, 0, 4, 2, 1, 181, 84, 39, 89, 40, 18, 9073, 3963, 1731, 94, 40, 17, 29, 12, 5, 5401, 2190, 888, 16001, 6350, 2520, 324, 126, 49, 55, 21, 8, 64, 24, 9, 361, 133, 49

Offset: 1

Author

Xianwen Wang, May 29 2025

Keywords

• nonn
• tabf

Comments

Given n, n!=k^3, there are infinitely many solutions, and all other solutions can be derived from the minimal solution pair by a recurrence relation. See Wolfe, pages 359-369.

Examples

			For n=5, the minimal positive solution is (41, 24, 14), so a(13)=41, a(14)=24, a(15)=14.
The array begins:
    1,  0,  0,
    1,  1,  1,
    4,  3,  2,
    5,  3,  2,
   41, 24, 14,
  109, 60, 33,
  ...
		

References

• Clyde Lynne Earle Wolfe, On the Indeterminate Cubic Equation X^3 + Dy^3 + D^2z^3 - 3Dxyz, University of California Press, 1923, pp. 359-369.

Links

• Xianwen Wang, Table of n, a(n) for n = 1..6000

Extensions

Name edited by Michel Marcus, Jun 03 2025