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%I #63 Jun 03 2022 07:41:16
%S 1,3,4,10,16,36,64,136,256,528,1024,2080,4096,8256,16384,32896,65536,
%T 131328,262144,524800,1048576,2098176,4194304,8390656,16777216,
%U 33558528,67108864,134225920,268435456,536887296,1073741824,2147516416,4294967296,8590000128
%N Number of undirected walks of length n+1 on an oriented triangle, visiting n+2 vertices, with n "corners"; the symmetry group is C3. Walks are not self-avoiding.
%C For a way to obtain this sequence from symmetry in quilts, see the Tom Young web page.
%C Also arises from the enumeration of based polyhedra with exactly two triangular faces [Rademacher]. - _N. J. A. Sloane_, Apr 24 2020
%C a(n-1) is the number of linear oriented trees with n arcs (n+1 nodes). - _R. J. Mathar_, Jun 09 2020
%H Vincenzo Librandi, <a href="/A051437/b051437.txt">Table of n, a(n) for n = 0..1000</a>
%H Hans Rademacher, <a href="https://doi.org/10.1215/ijm/1256068140">On the number of certain types of polyhedra</a>, Illinois Journal of Mathematics 9.3 (1965): 361-380. Reprinted in Coll. Papers, Vol II, MIT Press, 1974, pp. 544-564. See Theorem 7, Eq. 14.1.
%H A. Yajima, <a href="https://www.jstage.jst.go.jp/article/bcsj/87/11/87_20140204/_pdf">How to calculate the number of stereoisomers of inositol-homologs</a>, Bull. Chem. Soc. Jpn. 2014, 87, 1260-1264 | doi:10.1246/bcsj.20140204. See Tables 1 and 2 (and text). - _N. J. A. Sloane_, Mar 26 2015
%H Joseph D. Yelk, <a href="https://search.proquest.com/docview/2333655328">Molecular Dynamics Investigations of Duplex Columnar Liquid Crystal Phases of Nucleoside Triphosphates</a>, Ph. D. thesis, Northwestern University (2008).
%H Tom Young, <a href="http://www2.informns.k12.mn.us/schoolties/class/homeroom1.cfm?class=tyoung1">Math Research Quilt Pattern Symmetry</a> [Broken link]
%H Tom Young, <a href="/A051437/a051437.pdf">Unique symmetrical triangle quilt patterns along the diagonal of an nxn square</a> (An archived copy from the above page)
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,2,-4).
%F a(2n+1) = A007582(n+1). a(2n) = A000302(n).
%F a(n) = A000079(n) + A077957(n-1). - _Antti Karttunen_, Dec 29 2013
%F From _Paul Barry_, Apr 28 2004: (Start)
%F Binomial transform is 3^n + Pell(n) = (A000244(n) + A000129(n)).
%F G.f.: (1+x-4*x^2)/((1-2*x)(1-2*x^2));
%F a(n) = 2^n + 2^(n/2)*(1-(-1)^n)/(2*sqrt(2)). (End)
%F a(n) = 2*a(n-1) + 2*a(n-2) - 4*a(n-3); a(0)=1, a(1)=3, a(2)=4. - _Harvey P. Dale_, Jun 06 2011
%F a(n) = 2*a(n-2) + 2^(n-1), a(0)=1, a(1)=3. - _Yuchun Ji_, Aug 12 2020
%F E.g.f.: cosh(2*x) + sinh(2*x) + sinh(sqrt(2)*x)/sqrt(2). - _Stefano Spezia_, Jun 03 2022
%e For n=3 the walks visit vertices 1212, 1213, 1232, 1231.
%t LinearRecurrence[{2,2,-4},{1,3,4},50] (* or *) CoefficientList[ Series[ (1+x-4x^2)/((1-2x)(1-2x^2)),{x,0,50}],x] (* _Harvey P. Dale_, Jun 06 2011 *)
%o (Scheme) (define (A051437 n) (if (zero? n) 1 (+ (A000079 n) (A077957 (- n 1))))) ;; _Antti Karttunen_, Dec 29 2013
%Y Cf. A000079, A000129, A000244, A000302, A005418, A007582, A077957, A193231.
%K nonn,walk,nice,easy
%O 0,2
%A _Colin Mallows_
%E More terms from _Harvey P. Dale_, Jun 06 2011
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