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

Author

Colin Mallows

Keywords

• nonn
• walk
• nice
• easy

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

Examples

			For n=3 the walks visit vertices 1212, 1213, 1232, 1231.
		

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).

Crossrefs

Cf. A000079, A000129, A000244, A000302, A005418, A007582, A077957, A193231.

Programs

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 *)

Scheme

(define (A051437 n) (if (zero? n) 1 (+ (A000079 n) (A077957 (- n 1))))) ;; Antti Karttunen, Dec 29 2013

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

Extensions

More terms from Harvey P. Dale, Jun 06 2011