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 A006359 Number of distributive lattices; also number of paths with n turns when light is reflected from 6 glass plates. (Formerly M4148) 18
 1, 6, 21, 91, 371, 1547, 6405, 26585, 110254, 457379, 1897214, 7869927, 32645269, 135416457, 561722840, 2330091144, 9665485440, 40093544735, 166312629795, 689883899612, 2861717685450, 11870733787751, 49241167758705, 204258021937291, 847285745315256 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Let M denotes the 6 X 6 matrix = row by row (1,1,1,1,1,1)(1,1,1,1,1,0)(1,1,1,1,0,0)(1,1,1,0,0,0)(1,1,0,0,0,0)(1,0,0,0,0,0) and A(n) the vector (x(n),y(n),z(n),t(n),u(n),v(n)) = M^n*A where A is the vector (1,1,1,1,1,1) then a(n) = x(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. 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. 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. G. Kreweras, Les prĂ©ordres totaux compatibles avec un ordre partiel, Math. Sci. Humaines No. 53 (1976), 5-30. G. Kreweras, Les prĂ©ordres totaux compatibles avec un ordre partiel, Math. Sci. Humaines No. 53 (1976), 5-30. (Annotated scanned copy) Index entries for linear recurrences with constant coefficients, signature (3,6,-4,-5,1,1). FORMULA G.f.: -(z^4 + z^3 - 3z^2 - 2z + 1) / (-1 + 3z + 6z^2 - 4z^3 - 5z^4 + z^5 + z^6). - M. Goebel (manfredg(AT)ICSI.Berkeley.EDU) Jul 26 1997 a(n) = 3*a(n-1) + 6*a(n-2) - 4*a(n-3) - 5*a(n-4) + a(n-5) + a(n-6). a(n) is asymptotic to z(6)*w(6)^n where w(6) = (1/2)/cos(6*Pi/13) and z(6) is the root 1 < x < 2 of P(6, X) = -1 - 91*X + 2366*X^2 + 26364*X^3 - 142805*X^4 - 371293*X^5 + 371293*X^6 - Benoit Cloitre, Oct 16 2002 G.f.: A(x) = (1 + 3*x - 3*x^2 - 4*x^3 + x^4 + x^5)/(1 - 3*x - 6*x^2 + 4*x^3 + 5*x^4 - x^5 - x^6). - Paul D. Hanna, Feb 06 2006 G.f.: 1/(-x-1/(-x-1/(-x-1/(-x-1/(-x-1/(-x-1)))))). - Paul Barry, Mar 24 2010 MAPLE A=seq(a.j, j=0..5):grammar1:=[Q5, { seq(Q.i=Union(Epsilon, seq(Prod(a.j, Q.j), j=5-i..5)), i=0..5), seq(a.j=Z, j=0..5) }, unlabeled]: seq(count(grammar1, size=j), j=0..22); # Zerinvary Lajos, Mar 09 2007 MATHEMATICA LinearRecurrence[{3, 6, -4, -5, 1, 1}, {1, 6, 21, 91, 371, 1547}, 30] (* Harvey P. Dale, Sep 03 2016 *) PROG (PARI) k=5; M(k)=matrix(k, k, i, j, if(1-sign(i+j-k), 0, 1)); v(k)=vector(k, i, 1); a(n)=vecmax(v(k)*M(k)^n) (PARI) {a(n)=local(p=6); polcoeff(sum(k=0, p-1, (-1)^((k+1)\2)*binomial((p+k-1)\2, k)* (-x)^k)/sum(k=0, p, (-1)^((k+1)\2)*binomial((p+k)\2, k)*x^k+x*O(x^n)), n)} // Paul D. Hanna, Feb 06 2006 CROSSREFS Cf. A000217, A000330, A050446, A050447, A006356, A006357, A006358. See also A025030, A030112-A030116. Sequence in context: A005498 A002222 A290355 * A001553 A009247 A093774 Adjacent sequences:  A006356 A006357 A006358 * A006360 A006361 A006362 KEYWORD nonn,easy AUTHOR EXTENSIONS Alternative description from Jacques Haubrich (jhaubrich(AT)freeler.nl) More terms from James A. Sellers, Dec 24 1999 STATUS approved

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Last modified October 14 00:10 EDT 2019. Contains 327990 sequences. (Running on oeis4.)