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A160572 Elements of A160444, pairs of consecutive entries swapped. 0
1, 0, 1, 1, 4, 2, 10, 6, 28, 16, 76, 44, 208, 120, 568, 328, 1552, 896, 4240, 2448, 11584, 6688, 31648, 18272, 86464, 49920, 236224, 136384, 645376, 372608, 1763200, 1017984, 4817152, 2781184, 13160704, 7598336, 35955712, 20759040, 98232832 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

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 pairwise 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 nonzero integer starting vectors. - Hagen von Eitzen, May 22 2009

LINKS

Table of n, a(n) for n=1..39.

W. Limbrunner, Das Quadrat, ein Wunder der Geometrie (in German)

Index entries for linear recurrences with constant coefficients, signature (0, 2, 0, 2).

FORMULA

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, 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. - Hagen von Eitzen, May 22 2009

EXAMPLE

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,..

CROSSREFS

Cf. A160444, A002605 (bisection), A026150 (bisection).

Sequence in context: A283942 A266418 A193422 * A213500 A213584 A213587

Adjacent sequences:  A160569 A160570 A160571 * A160573 A160574 A160575

KEYWORD

nonn,easy

AUTHOR

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

EXTENSIONS

Edited by R. J. Mathar, May 22 2009

STATUS

approved

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Last modified January 19 14:53 EST 2020. Contains 331049 sequences. (Running on oeis4.)