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 A006358 Number of distributive lattices; also number of paths with n turns when light is reflected from 5 glass plates. (Formerly M3862) 10
 1, 5, 15, 55, 190, 671, 2353, 8272, 29056, 102091, 358671, 1260143, 4427294, 15554592, 54648506, 191998646, 674555937, 2369942427, 8326406594, 29253473175, 102777312308, 361091343583, 1268635610806, 4457144547354 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Let M denotes the 5 X 5 matrix = row by row (1,1,1,1,1)(1,1,1,1,0)(1,1,1,0,0)(1,1,0,0,0)(1,0,0,0,0) and A(n) the vector (x(n),y(n),z(n),t(n),u(n))=M^n*A where A is the vector (1,1,1,1,1); then a(n)=y(n). - Benoit Cloitre, Apr 02 2002 REFERENCES J. Berman and P. Koehler, Cardinalities of finite distributive lattices, Mitteilungen aus dem Mathematischen Seminar Giessen, 121 (1976), 103-124. S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (see p. 120). J. Haubrich, Multinacci Rijen [Multinacci sequences], Euclides (Netherlands), Vol. 74, Issue 4, 1998, pp. 131-133. N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS T. D. Noe, Table of n, a(n) for n=0..200 J. Berman and P. Koehler, Cardinalities of finite distributive lattices, Mitteilungen aus dem Mathematischen Seminar Giessen, 121 (1976), 103-124. [Annotated scanned copy] Emma L. L. Gao, Sergey Kitaev, Philip B. Zhang, Pattern-avoiding alternating words, preprint, 2015. Manfred Goebel, Rewriting Techniques and Degree Bounds for Higher Order Symmetric Polynomials, Applicable Algebra in Engineering, Communication and Computing (AAECC), Volume 9, Issue 6 (1999), 559-573. G. Kreweras, Les preordres totaux compatibles avec un ordre partiel, Math. Sci. Humaines No. 53 (1976), 5-30. Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992. Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992. Index entries for linear recurrences with constant coefficients, signature (3,3,-4,-1,1). FORMULA a(n) = 3*a(n-1) + 3*a(n-2) - 4*a(n-3) - a(n-4) + a(n-5). a(n) is asymptotic to z(5)*w(5)^n where w(5)=(1/2)/cos(5*Pi/11) and z(5) is the root 1

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