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.

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

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

E.g.f.: cosh(2*x) + sinh(2*x) + sinh(sqrt(2)*x)/sqrt(2). - Stefano Spezia, Jun 03 2022

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, A000129, A000244, A000302, A005418, A007582, A077957, A193231.

Sequence in context: A281903 A093512 A081160 * A224073 A034774 A342536

Adjacent sequences: A051434 A051435 A051436 * A051438 A051439 A051440

Keyword

nonn,walk,nice,easy

Author

Colin Mallows

Extensions

More terms from Harvey P. Dale, Jun 06 2011

Status

approved