A226700 Solutions y/(3*5*13) of the Pell equation x^2 - 61*y^2 = +4.

0, 1, 1523, 2319528, 3532639621, 5380207823255, 8194052982177744, 12479537311648880857, 19006327131588263367467, 28946623741871613459771384, 44085688952543335710968450365, 67142475328099758416191490134511

Offset

0,3

Comments

y' = a(n) and x = b(n) := A226699(n) are the nonnegative solutions of x^2 - 61*(3*5*13*y')^2 = +4. This is x^2 - D*y'^2 = +4 with D = (61*(3*5*13)^2 = 61*195^2 = 2319525.

References

T. Nagell, Introduction to Number Theory, Chelsea Publishing Company, New York, 1964, ch. Vi, 58., p. 204-212.

Links

Table of n, a(n) for n=0..11.

Index entries for sequences related to Chebyshev polynomials.

Index entries for linear recurrences with constant coefficients, signature (1523,-1).

Formula

a(n) = S(n-1,1523), n >= 0, with the Chebyshev S-polynomials (A049310), with S(-1, 0) = 0.

O.g.f. x/(1- 1523*x + x^2).

a(n) = 1523*a(n-1) - a(n-2), n >= 1, a(-1) = -1, a(0) = 0.

Mathematica

LinearRecurrence[{1523, -1}, {0, 1}, 20] (* Harvey P. Dale, Apr 24 2023 *)

Crossrefs

Cf. A226699.

Sequence in context: A217387 A031717 A236880 * A226701 A235170 A252414

Adjacent sequences: A226697 A226698 A226699 * A226701 A226702 A226703

Keyword

nonn,easy

Author

Wolfdieter Lang, Jun 27 2013

Status

approved