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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. 4
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.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

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

Tom Young, Math Research Quilt Pattern Symmetry

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

n=2m: a(n)=2^(n-1)+2^((n-2)/2); n=2m+1: a(n)=2^(n-1). - [Note: This formula is for starting offset=1.]

a(n) = A000079(n) + A077957(n-1). - Antti Karttunen, Dec 29 2013

Binomial transform is 3^n+Pell(n) (A000244(n)+A000129(n)). G.f.: (1+x-4x^2)/((1-2x)(1-2x^2)); a(n)=2^n+2^(n/2)(1-(-1)^n)/(2sqrt(2)). - Paul Barry, Apr 28 2004

a(0)=1, a(1)=3, a(2)=4, a(n)=2*a(n-1)+2*a(n-2)-4*a(n-3). - Harvey P. Dale, Jun 06 2011

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 A172416

Adjacent sequences:  A051434 A051435 A051436 * A051438 A051439 A051440

KEYWORD

nonn,nice,easy

AUTHOR

Colin Mallows

EXTENSIONS

More terms from Harvey P. Dale, Jun 06 2011

STATUS

approved

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Last modified August 19 14:06 EDT 2017. Contains 290808 sequences.