A126866 a(n) = 13*a(n-1) - a(n-2).

1, 14, 181, 2339, 30226, 390599, 5047561, 65227694, 842912461, 10892634299, 140761333426, 1819004700239, 23506299769681, 303762892305614, 3925411300203301, 50726584010337299, 655520180834181586, 8471035766834023319, 109467944788008121561

Offset: 0

Author

Diego A. Penta (diego(AT)alum.mit.edu), Mar 15 2007

Keywords

• easy
• nonn

Comments

Nonnegative x values in solutions to the Diophantine equation 11*x^2 - 15*y^2 = -4. The corresponding y values are in A085260. Note that a(n+1)^2 - a(n)*a(n+2) = 15. - Klaus Purath, Mar 21 2025

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..899
• K. Andersen, L. Carbone, and D. Penta, Kac-Moody Fibonacci sequences, hyperbolic golden ratios, and real quadratic fields, Journal of Number Theory and Combinatorics, Vol 2, No. 3 pp 245-278, 2011. See Section 9.
• Alex Fink, Richard K. Guy, and Mark Krusemeyer, Partitions with parts occurring at most thrice, Contributions to Discrete Mathematics, Vol 3, No 2 (2008), pp. 76-114. See Section 13.
• Index entries for linear recurrences with constant coefficients, signature (13,-1).

Crossrefs

Cf. A002878, A001834, A030221, A002315.

Programs

Mathematica

LinearRecurrence[{13,-1},{1,14},30] (* Harvey P. Dale, Mar 28 2013 *)

Sage

[(lucas_number2(n,13,1)-lucas_number2(n-1,13,1))/11 for n in range(1, 16)] # Zerinvary Lajos, Nov 10 2009

Formula

a(n) = 13*a(n-1) - a(n-2); a(0)=1, a(1)=14.

G.f.: (x+1)/(x^2-13*x+1). - Harvey P. Dale, Mar 28 2013

Extensions

Corrected and extended by Harvey P. Dale, Mar 28 2013