cp's OEIS Frontend

A092812 Number of closed walks of length 2*n on the 4-cube.

%I A092812 #52 Nov 10 2024 21:45:29
%S A092812 1,4,40,544,8320,131584,2099200,33562624,536903680,8590065664,
%T A092812 137439477760,2199025352704,35184380477440,562949986975744,
%U A092812 9007199388958720,144115188612726784,2305843011361177600
%N A092812 Number of closed walks of length 2*n on the 4-cube.
%C A092812 With interpolated zeros this has a(n) = (6*0^n + 4^n + (-4)^n + 4*2^n + 4*(-2)^n)/16 and counts closed walks of length n at a vertex of the 4-cube. [Typo corrected by _Alexander R. Povolotsky_, May 26 2008]
%C A092812 Also, cogrowth sequence of the 16-element group C2^4. - _Sean A. Irvine_, Nov 10 2024
%H A092812 Vincenzo Librandi, <a href="/A092812/b092812.txt">Table of n, a(n) for n = 0..200</a>
%H A092812 G. R. Franssens, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Franssens/franssens13.html">On a number pyramid related to the binomial, Deleham, Eulerian, MacMahon and Stirling number triangles</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.4.1.
%H A092812 Katarzyna Grygiel, Pawel M. Idziak and Marek Zaionc, <a href="http://arxiv.org/abs/1112.0643">How big is BCI fragment of BCK logic</a>, arXiv preprint arXiv:1112.0643 [cs.LO], 2011. [From _N. J. A. Sloane_, Feb 21 2012]
%H A092812 L. Reyzin, Mathoverflow, <a href="https://mathoverflow.net/questions/71736/number-of-closed-walks-on-an-n-cube">Number of closed walks on an n-cube</a>.
%H A092812 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (20,-64).
%F A092812 G.f.: (1-16*x+24*x^2)/((1-4*x)*(1-16*x)).
%F A092812 a(n) = 3*0^n/8 + 16^n/8 + 4^n/2.
%F A092812 From _Peter Bala_, Nov 13 2006: (Start)
%F A092812 E.g.f.: cosh^4(x).
%F A092812 O.g.f.: 1/(1-4*1*x/(1-3*2*x/(1-2*3*x/(1-1*4*x)))) (continued fraction). (End)
%F A092812 (-1)^n*a(n) = Sum_{k=0..n} A086872(n,k)*(-5)^(n-k). - _Philippe Deléham_, Aug 17 2007
%F A092812 a(n) = 20*a(n-1) - 64*a(n-2); a(0) = 1, a(1) = 4, a(2) = 40. - _Harvey P. Dale_, Aug 23 2011
%F A092812 a(n) = 4*A026244(n-1), n > 0. - _R. J. Mathar_, Oct 24 2014
%F A092812 a(n) = (1/2^4)*Sum_{j = 0..4} binomial(4, j)*(4 - 2*j)^(2*n). See Reyzin link. - _Peter Bala_, Jun 03 2019
%t A092812 CoefficientList[Series[(1-16x+24x^2)/((1-4x)(1-16x)),{x,0,30}],x] (* or *) Join[{1},LinearRecurrence[{20,-64},{4,40},30]] (* _Harvey P. Dale_, Aug 23 2011 *)
%o A092812 (Magma) [3*0^n/8+16^n/8+4^n/2: n in [0..30]]; // _Vincenzo Librandi_, May 31 2011
%Y A092812 Essentially the same as A075878.
%Y A092812 Cf. A026244, A081294, A054879, A121822.
%K A092812 easy,nonn
%O A092812 0,2
%A A092812 _Paul Barry_, Mar 11 2004
%E A092812 Title improved by _Sean A. Irvine_ at the suggestion of _Peter Bala_, Jun 04 2019