A041612 Numerators of continued fraction convergents to sqrt(325).

18, 649, 23382, 842401, 30349818, 1093435849, 39394040382, 1419278889601, 51133434066018, 1842222905266249, 66371158023650982, 2391203911756701601, 86149711981264908618, 3103780835237293411849, 111822259780523827735182, 4028705132934095091878401

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• cofr
• frac
• easy

Comments

a(2*n) and b(2*n) = A041613(2*n) give all (positive integer) solutions to the Pell equation a^2 - 13*b^2 = -1. a(2*n+1) and b(2*n+1) = A041613(2*n+1) give all (positive integer) solutions to the Pell equation a^2 - 13*b^2 = 1. - Robert FERREOL, Oct 09 2024

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Tanya Khovanova, Recursive Sequences
• Index entries for linear recurrences with constant coefficients, signature (36,1).

Crossrefs

Cf. A040306 (continued fraction), A041613 (denominators), A295330.

Programs

Mathematica

Numerator[Convergents[Sqrt[325], 30]] (* Vincenzo Librandi, Nov 04 2013 *)

Formula

From Philippe Deléham, Nov 23 2008: (Start)

a(n) = 36*a(n-1) + a(n-2), n > 1; a(0)=18, a(1)=649.

G.f.: (18+x)/(1-36*x-x^2). (End)

a(n) = ((18 + 5*sqrt(13))^(n+1) + (18 - 5*sqrt(13))^(n+1))/2. - Robert FERREOL, Oct 09 2024

Extensions

Additional term from Colin Barker, Nov 09 2013