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A160572 Elements of A160444, pairs of consecutive entries swapped.

%I A160572 #19 Feb 04 2025 23:38:09
%S A160572 1,0,1,1,4,2,10,6,28,16,76,44,208,120,568,328,1552,896,4240,2448,
%T A160572 11584,6688,31648,18272,86464,49920,236224,136384,645376,372608,
%U A160572 1763200,1017984,4817152,2781184,13160704,7598336,35955712,20759040,98232832
%N A160572 Elements of A160444, pairs of consecutive entries swapped.
%C A160572 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.
%C A160572 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.
%C A160572 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
%H A160572 W. Limbrunner, <a href="http://bewusstsein.xobor.de/t229f67-Das-Quadrat-ein-Wunder-der-Geometrie.html">Das Quadrat, ein Wunder der Geometrie</a> (in German)
%H A160572 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0, 2, 0, 2).
%F A160572 a(2*n+1)=A160444(2*n+2). a(2*n+2)=A160444(2*n+1).
%F A160572 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
%F A160572 a(1)=1, a(2)=0, and for n>=1: a(2*n+1) = a(2*n-1)+3*a(2*n), a(2*n+2) = a(2*n+1)+a(2*n). Or: Let c1 = 1+sqrt(3), c2 = 1-sqrt(3). Then a(2*n+1) = (c1^n + c2^n)/2, a(2*n+2) = (c1^n - c2^n)/(2*sqrt(3)) for n >= 0. - _Hagen von Eitzen_, May 22 2009
%e A160572 k=2: 1,0,1,1,3,2,7,5,17,12,41,29,99,70,239,169,577,408,1393,985
%e A160572 k=3: 1,0,1,1,4,2,10,6,28,16,76,44,208,120,568,328,1552... (here)
%e A160572 k=4: 1,0,1,1,5,2,13,7,41,20,121,61,365,182,1093,547,3281,..
%e A160572 k=5: 1,0,1,1,6,2,16,8,56,24,176,80,576,256,1856,832,6016,2688,..
%e A160572 k=6: 1,0,1,1,7,2,19,9,73,28,241,101,847,342,2899,1189,..
%e A160572 k=7: 1,0,1,1,8,2,22,10,92,32,316,124,1184,440,4264,1624,..
%e A160572 k=8: 1,0,1,1,9,2,25,11,113,36,401,149,1593,550,5993,2143,..
%e A160572 k=9: 1,0,1,1,10,2,28,12,136,40,496,176,2080,672,8128,2752,..
%e A160572 k=10: 1,0,1,1,11,2,31,13,161,44,601,205,2651,806,10711,3457,..
%Y A160572 Cf. A160444, A002605 (bisection), A026150 (bisection).
%K A160572 nonn,easy
%O A160572 1,5
%A A160572 Willibald Limbrunner (w.limbrunner(AT)gmx.de), May 20 2009
%E A160572 Edited by _R. J. Mathar_, May 22 2009