A160572 - OEIS

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A160572

Elements of A160444, pairs of consecutive entries swapped.

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; 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 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]`
	FORMULA	`a(2n+1)=A160444(2n+2). a(2n+2)=A160444(2n+1).` `G.f.: -x(1-x^2+x^3)/(-1+2x^2+(k-1)x^4). a(n)=2a(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)+3a(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]`
	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.` `Sequence in context: A191725 A198326 A193422 * A066579 A117821 A185732` `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 (mathar(AT)strw.leidenuniv.nl), May 22 2009`

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Last modified February 17 08:44 EST 2012. Contains 205998 sequences.