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 A051437 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. 8
 1, 3, 4, 10, 16, 36, 64, 136, 256, 528, 1024, 2080, 4096, 8256, 16384, 32896, 65536, 131328, 262144, 524800, 1048576, 2098176, 4194304, 8390656, 16777216, 33558528, 67108864, 134225920, 268435456, 536887296, 1073741824, 2147516416, 4294967296, 8590000128 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS For a way to obtain this sequence from symmetry in quilts, see the Tom Young web page. Also arises from the enumeration of based polyhedra with exactly two triangular faces [Rademacher]. - N. J. A. Sloane, Apr 24 2020 a(n-1) is the number of linear oriented trees with n arcs (n+1 nodes). - R. J. Mathar, Jun 09 2020 LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..1000 Hans Rademacher, On the number of certain types of polyhedra, 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. A. Yajima, How to calculate the number of stereoisomers of inositol-homologs, 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 Joseph D. Yelk, Molecular Dynamics Investigations of Duplex Columnar Liquid Crystal Phases of Nucleoside Triphosphates, Ph. D. thesis, Northwestern University (2008). Tom Young, Math Research Quilt Pattern Symmetry [Broken link] Tom Young, Unique symmetrical triangle quilt patterns along the diagonal of an nxn square (An archived copy from the above page) Index entries for linear recurrences with constant coefficients, signature (2,2,-4). FORMULA a(2n+1) = A007582(n+1). a(2n) = A000302(n). a(n) = A000079(n) + A077957(n-1). - Antti Karttunen, Dec 29 2013 From Paul Barry, Apr 28 2004: (Start) Binomial transform is 3^n + Pell(n)*(A000244(n) + A000129(n)). G.f.: (1+x-4*x^2)/((1-2*x)(1-2*x^2)); a(n) = 2^n + 2^(n/2)*(1-(-1)^n)/(2*sqrt(2)). (End) 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 a(n) = 2*a(n-2) + 2^(n-1), a(0)=1, a(1)=3. - Yuchun Ji, Aug 12 2020 EXAMPLE For n=3 the walks visit vertices 1212, 1213, 1232, 1231. MATHEMATICA 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 *) PROG (Scheme) (define (A051437 n) (if (zero? n) 1 (+ (A000079 n) (A077957 (- n 1))))) ;; Antti Karttunen, Dec 29 2013 CROSSREFS Cf. A000079, A077957, A005418, A193231. Sequence in context: A281903 A093512 A081160 * A224073 A034774 A342536 Adjacent sequences:  A051434 A051435 A051436 * A051438 A051439 A051440 KEYWORD nonn,walk,nice,easy AUTHOR EXTENSIONS More terms from Harvey P. Dale, Jun 06 2011 STATUS approved

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Last modified April 17 06:46 EDT 2021. Contains 343059 sequences. (Running on oeis4.)