1,5

The case k=3 of a family of sequences defined by a(1)=1, a(2)=0,

a(2n+1)=a(2n-1)+k*a(2n), a(2n+2)=a(2n)+a(2n-1), each congruent to

one of the sequences mentioned in A160444 by pair-wise interchanges. The case

k=2 is covered by swapping pairs in A002965.

Each of the two subsequences b(n) obtained by bisection has a limiting ratio

b(n+1)/b(n)=1+sqrt(k) by Binet's Formula. In a logarithmic plot of the

sequence a(n) one therefore sees a staircase, the two edges at each step alternately

marked by one of the two subsequences.

Matrix M = [[1 3] [1 1]] is iterated with starting vector [1 0]^T. Since M has eigenvectors [ +-sqrt(3) 1]^T with eigenvalues 1 +- sqrt(3), we have lim xn/yn = 1+sqrt(3) for all non-zero integer starting vectors. [From Hagen von Eitzen (math(AT)von-eitzen.de), May 22 2009]

a(2n+1)=A160444(2n+2). a(2n+2)=A160444(2n+1).

G.f.: -x*(1-x^2+x^3)/(-1+2*x^2+(k-1)*x^4). a(n)=2*a(n-2)+(k-1)*a(n-4) at k=3. [R. J. Mathar (mathar(AT)strw.leidenuniv.nl), May 22 2009]

a(1)=1, a(2)=0, and for n>=1: a(2n+1) = a(2n-1)+3*a(2n), a(2n+2) = a(2n+1)+a(2n). Or: Let c1 = 1+sqrt(3), c2 = 1-sqrt(3). Then a(2n+1) = (c1^n + c2^n)/2, a(2n+2)) = (c1^n - c2^n)/(2*sqrt(3)) for n >= 0. [From Hagen von Eitzen (math(AT)von-eitzen.de), May 22 2009]

k=2: 1,0,1,1,3,2,7,5,17,12,41,29,99,70,239,169,577,408,1393,985

k=3: 1,0,1,1,4,2,10,6,28,16,76,44,208,120,568,328,1552... (here)

k=4: 1,0,1,1,5,2,13,7,41,20,121,61,365,182,1093,547,3281,..

k=5: 1,0,1,1,6,2,16,8,56,24,176,80,576,256,1856,832,6016,2688,..

k=6: 1,0,1,1,7,2,19,9,73,28,241,101,847,342,2899,1189,..

k=7: 1,0,1,1,8,2,22,10,92,32,316,124,1184,440,4264,1624,..

k=8: 1,0,1,1,9,2,25,11,113,36,401,149,1593,550,5993,2143,..

k=9: 1,0,1,1,10,2,28,12,136,40,496,176,2080,672,8128,2752,..

k=10: 1,0,1,1,11,2,31,13,161,44,601,205,2651,806,10711,3457,..

Cf. A160444.

Sequence in context: A130273 A016516 A138569 this_sequence A066579 A117821 A121794

Adjacent sequences: A160569 A160570 A160571 this_sequence A160573 A160574 A160575

nonn,easy

Willibald Limbrunner (w.limbrunner(AT)gmx.de), May 20 2009

Edited by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), May 22 2009