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A006357
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Number of distributive lattices; also number of paths with n turns when light is reflected from 4 glass plates.
(Formerly M3396)
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16
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1, 4, 10, 30, 85, 246, 707, 2037, 5864, 16886, 48620, 139997, 403104, 1160693, 3342081, 9623140, 27708726, 79784098, 229729153, 661478734, 1904652103, 5484227157, 15791202736, 45468956106, 130922641160, 376976720745, 1085461206128, 3125460977225
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OFFSET
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0,2
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COMMENTS
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Let M denotes the 4 X 4 matrix = row by row (1,1,1,1)(1,1,1,0)(1,1,0,0)(1,0,0,0) and A(n) the vector (x(n),y(n),z(n),t(n))=M^n*A where A is the vector (1,1,1,1) then a(n)=x(n). - Benoit Cloitre, Apr 02 2002
In general, the g.f. for p glass plates is A(x) = F_{p-1}(-x)/F_p(x) where F_p(x) = Sum_{k=0,p} (-1)^[(k+1)/2]*C([(p+k)/2],k)*x^k. - Paul D. Hanna, Feb 06 2006
a(n)/a(n-1) tends to 2.879385..., the longest diagonal of a nonagon with edge 1; or: sin(4*Pi/9)/sin(Pi/9). The sequence is the INVERT transform of (1, 3, 3, 5, 8, 13, 21, 34, 55, 89, 144, ...). - Gary W. Adamson, Jul 16 2015
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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.
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).
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LINKS
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FORMULA
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G.f.: (1 + 2*x - x^2 - x^3)/( (1 +x)*(1 -3*x +x^3) ). - Simon Plouffe in his 1992 dissertation
a(n) = 2*a(n-1) + 3*a(n-2) - a(n-3) - a(n-4).
a(n) is asymptotic to z(4)*w(4)^n where w(4) = (1/2)/cos(4*Pi/9) and z(4) is the root 1 < x < 2 of P(4, X) = 1 + 27*X - 324*X^2 + 243*X^3. - Benoit Cloitre, Oct 16 2002
G.f.: 1/(-x-1/(-x-1/(-x-1/(-x-1)))). - Paul Barry, Mar 24 2010
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MATHEMATICA
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LinearRecurrence[{2, 3, -1, -1}, {1, 4, 10, 30}, 30] (* Harvey P. Dale, Nov 18 2013 *)
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PROG
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(PARI) a(n)=local(p=4); 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
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CROSSREFS
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KEYWORD
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nonn,nice,easy
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AUTHOR
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EXTENSIONS
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Recurrence, alternative description from Jacques Haubrich (jhaubrich(AT)freeler.nl)
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STATUS
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approved
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