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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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17
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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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 (benoit7848c(AT)orange.fr), Apr 02 2002
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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.
G. Kreweras, Les preordres totaux compatibles avec un ordre partiel. Math. Sci. Humaines No. 53 (1976), 5-30.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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FORMULA
| G.f. from M. Goebel (manfredg(AT)ICSI.Berkeley.EDU) Jul 26 1997: -(z^4 + z^3 - 3z^2 - 2z + 1) / (-1 + 3z + 6z^2 - 4z^3 - 5z^4 + z^5 + z^6).
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 (benoit7848c(AT)orange.fr), 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 (pauldhanna(AT)juno.com), Feb 06 2006
G.f.: 1/(-x-1/(-x-1/(-x-1/(-x-1/(-x-1/(-x-1)))))). [From Paul Barry (pbarry(AT)wit.ie), Mar 24 2010]
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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 (zerinvarylajos(AT)yahoo.com), Mar 09 2007
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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 (pauldhanna(AT)juno.com), Feb 06 2006
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CROSSREFS
| Cf. A000217, A000330, A050446, A050447, A006356, A006357, A006358.
See also A025030, A030112-A030116.
Sequence in context: A137966 A005498 A002222 * A001553 A009247 A093774
Adjacent sequences: A006356 A006357 A006358 * A006360 A006361 A006362
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KEYWORD
| nonn,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| Alternative description from Jacques Haubrich (jhaubrich(AT)freeler.nl).
More terms from James A. Sellers (sellersj(AT)math.psu.edu), Dec 24 1999
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