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A006359
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Number of distributive lattices; also number of paths with n turns when light is reflected from 6 glass plates.
(Formerly M4148)
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18
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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
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OFFSET
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0,2
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COMMENTS
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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
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REFERENCES
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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).
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LINKS
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FORMULA
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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
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MAPLE
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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
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MATHEMATICA
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LinearRecurrence[{3, 6, -4, -5, 1, 1}, {1, 6, 21, 91, 371, 1547}, 30] (* Harvey P. Dale, Sep 03 2016 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,easy,changed
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AUTHOR
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EXTENSIONS
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Alternative description from Jacques Haubrich (jhaubrich(AT)freeler.nl)
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STATUS
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approved
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