0,3

The k-Fibonacci sequences for k=2..12 are A000129, A006190, A001076, A052918,

A005668, A054413, A041025, A099371, A041041, A049666, A041061. This here is

k=13. k=14 is A041085, k=16 A041113, k=18 A041145, k=20 A041181, k=22 A041221.

Sergio Falcon, Angel Plaza, The k-Fibonacci sequence and Pascal 2-triangle, Chaos, Solit. Fract. 33 (2007) 38-49.

O.g.f.: x/(1-kx-x^2). a(n)=k*a(n-1)+a(n-2). a(n-r)*a(n+r)-a(n)^2=(-1)^(n+1-r)*a(r)^2. a(n)=sum_{i=0..[(n-1)/2]} binomial(n,2i+1) k^(n-1-2i)(k^2+4)^i/2^(n-1), k=13.

((13+sqrt173)^n-(13-sqrt173)^n)/(2^n*sqrt173). Offset 1. a(3)=170. [From Al Hakanson (hawkuu(AT)gmail.com), Jan 12 2009]

F := proc(n, k) coeftayl( x/(1-k*x-x^2), x=0, n) ; end: for n from 0 to 20 do printf("%d, ", F(n, 13)) ; od:

a=0; lst={a}; s=0; Do[a=s-(a-1); AppendTo[lst, a]; s+=a*13, {n, 3*4!}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Oct 27 2009]

(Other) sage: [lucas_number1(n, 13, -1) for n in xrange(0, 18)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 29 2009]

Sequence in context: A057684 A053153 A167254 this_sequence A041314 A065544 A096719

Adjacent sequences: A140452 A140453 A140454 this_sequence A140456 A140457 A140458

easy,nonn,new

R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 22 2008