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
From Paul Barry, Apr 28 2004: (Start)
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
KEYWORD
nonn,walk,nice,easy
AUTHOR
EXTENSIONS
More terms from Harvey P. Dale, Jun 06 2011
STATUS
approved