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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 préordres totaux compatibles avec un ordre partiel, Math. Sci. Humaines No. 53 (1976), 5-30. G. Kreweras, Les préordres totaux compatibles avec un ordre partiel, Math. Sci. Humaines No. 53 (1976), 5-30. (Annotated scanned copy) 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 < x < 2 of P(5, X) = -1 + 55*X + 847*X^2 - 5324*X^3 - 14641*X^4 + 14641*X^5. - Benoit Cloitre, Oct 16 2002 G.f.: A(x) = (1 + 2*x - 3*x^2 - x^3 + x^4)/(1 - 3*x - 3*x^2 + 4*x^3 + x^4 - x^5). - Paul D. Hanna, Feb 06 2006 MAPLE A=seq(a.j, j=0..4):grammar1:=[Q4, { seq(Q.i=Union(Epsilon, seq(Prod(a.j, Q.j), j=4-i..4)), i=0..4), seq(a.j=Z, j=0..4) }, unlabeled]: seq(count(grammar1, size=j), j=0..23); # Zerinvary Lajos, Mar 09 2007 A006358:=-(z-1)*(z**3-3*z-1)/(-1+3*z+3*z**2-4*z**3-z**4+z**5); # conjectured by Simon Plouffe in his 1992 dissertation MATHEMATICA m = Table[ If[j <= 6-i, 1, 0], {i, 1, 5}, {j, 1, 5}] ; a[n_] := MatrixPower[m, n].Table[1, {5}]; Table[ a[n], {n, 0, 23}][[All, 1]] (* Jean-François Alcover, Dec 08 2011, after Benoit Cloitre *) LinearRecurrence[{3, 3, -4, -1, 1}, {1, 5, 15, 55, 190}, 30] (* Harvey P. Dale, Jun 16 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=5); 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)} CROSSREFS Cf. A000217, A000330, A050446, A050447. See also A006356-A006359, A025030, A030112-A030116. Cf. A038201 (5-wave sequence). Sequence in context: A002221 A007714 A123011 * A054108 A149585 A114947 Adjacent sequences:  A006355 A006356 A006357 * A006359 A006360 A006361 KEYWORD nonn,easy,nice AUTHOR EXTENSIONS Alternative description and formula from Jacques Haubrich (jhaubrich(AT)freeler.nl) More terms from James A. Sellers, Dec 24 1999 STATUS approved

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Last modified November 19 04:22 EST 2018. Contains 317333 sequences. (Running on oeis4.)