This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A160444 #25 Mar 09 2023 09:01:27
%S A160444 0,1,1,1,2,4,6,10,16,28,44,76,120,208,328,568,896,1552,2448,4240,6688,
%T A160444 11584,18272,31648,49920,86464,136384,236224,372608,645376,1017984,
%U A160444 1763200,2781184,4817152,7598336,13160704,20759040,35955712,56714752
%N A160444 Expansion of g.f.: x^2*(1 + x - x^2)/(1 - 2*x^2 - 2*x^4).
%C A160444 This sequence is the case k=3 of a family of sequences with recurrences a(2*n+1) = a(2*n) + a(2*n-1), a(2*n+2) = k*a(2*n-1) + a(2*n), a(1)=0, a(2)=1. Values of k, for k >= 0, are given by A057979 (k=0), A158780 (k=1), A002965 (k=2), this sequence (k=3). See "Family of sequences for k" link for other connected sequences.
%C A160444 It seems that the ratio of two successive numbers with even, or two successive numbers with odd, indices approaches sqrt(k) for these sequences as n-> infinity.
%C A160444 This algorithm can be found in a historical figure named "Villardsche Figur" of the 13th century. There you can see a geometrical interpretation.
%H A160444 G. C. Greubel, <a href="/A160444/b160444.txt">Table of n, a(n) for n = 1..1000</a>
%H A160444 W. Beinert, <a href="https://www.typolexikon.de/villardscher-teilungskanon/">Villardscher Teilungskanon</a>, Lexikon der Typographie
%H A160444 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 A160444 Willibald Limbrunner, <a href="/A160444/a160444.txt">Family of sequences for k</a>
%H A160444 M-T. Zenner, <a href="http://www.emis.de/journals/NNJ/Zenner.html">Villard de Honnecourt and Euclidean Geoometry</a>, Nexus Network Journal 4 (2002) 65-78.
%H A160444 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,2,0,2).
%F A160444 a(n) = 2*a(n-2) + 2*a(n-4).
%F A160444 a(2*n+1) = A002605(n).
%F A160444 a(2*n) = A026150(n-1).
%t A160444 LinearRecurrence[{0,2,0,2}, {0,1,1,1}, 40] (* _G. C. Greubel_, Feb 18 2023 *)
%o A160444 (Magma) I:=[0,1,1,1]; [n le 4 select I[n] else 2*(Self(n-2) +Self(n-4)): n in [1..40]]; // _G. C. Greubel_, Feb 18 2023
%o A160444 (SageMath)
%o A160444 @CachedFunction
%o A160444 def a(n): # a = A160444
%o A160444 if (n<5): return ((n+1)//3)
%o A160444 else: return 2*(a(n-2) + a(n-4))
%o A160444 [a(n) for n in range(1, 41)] # _G. C. Greubel_, Feb 18 2023
%Y A160444 Cf. A002532, A002533, A002534, A002535, A002605, A002965, A003665.
%Y A160444 Cf. A003683, A015518, A015519, A026150, A046717, A057979, A063727.
%Y A160444 Cf. A083098, A083099, A083100, A084057, A158780.
%K A160444 nonn,easy
%O A160444 1,5
%A A160444 Willibald Limbrunner (w.limbrunner(AT)gmx.de), May 14 2009
%E A160444 Edited by _R. J. Mathar_, May 14 2009