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

1,7

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.

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

Example

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,
  ...

Crossrefs

Sequence in context: A333669 A159846 A377512 * A071890 A167837 A222228

Adjacent sequences: A384437 A384438 A384439 * A384441 A384442 A384443

Keyword

nonn,tabf

Author

Xianwen Wang, May 29 2025

Extensions

Name edited by Michel Marcus, Jun 03 2025

Status

approved