A226701 Positive solutions x/(3*13) of the Pell equation x^2 - 61*y^2 = -4.

1, 1524, 2321051, 3534959149, 5383740462876, 8199433190000999, 12487731364631058601, 19018806668899912248324, 28965630069003201723138851, 44114635576285207324428221749, 67186561017052301751902458584876

Offset

0,2

Comments

The proper and improper positive solutions of the Pell equation x^2 - 61*y^2 = -4 are x = 39*a(n) and y = 5*A226702(n), n >= 1.

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..10.

Index entries for sequences related to Chebyshev polynomials.

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

Formula

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

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

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

Example

n=1: (39*1)^2 - 61*(5*1)^2 = -4,
n=2: (39*1524)^2 - 61*(5*1522)^2 = -4,
n=3: (39*2321051)^2 - 61*(5*2318005)^2 = -4.

Mathematica

CoefficientList[Series[(1 + x)/(1 - 1523*x + x^2), {x, 0, 10}], x] (* Wesley Ivan Hurt, Jan 24 2017 *)

Crossrefs

Cf. A049310, A226702.

Sequence in context: A031717 A236880 A226700 * A235170 A252414 A333721

Adjacent sequences: A226698 A226699 A226700 * A226702 A226703 A226704

Keyword

nonn,easy

Author

Wolfdieter Lang, Jun 27 2013

Status

approved