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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
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
Marcin Gąsiorek, Structure of non-negative posets of Dynkin type A_n, arXiv:2205.15032 [math.CO], 2022-2023. See p. 18.
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.
Arata Yajima, How to calculate the number of stereoisomers of inositol homologs, Bull. Chem. Soc. Jpn. 2014, 87, 1260-1264. See Tables 1 and 2 (and text). - N. J. A. Sloane, Mar 26 2015
Tom Young, Math Research Quilt Pattern Symmetry [Broken link]
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
KEYWORD
nonn,walk,nice,easy
AUTHOR
EXTENSIONS
More terms from Harvey P. Dale, Jun 06 2011
STATUS
approved