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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.

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).

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.

Index entries for linear recurrences with constant coefficients, signature (3,6,-4,-5,1,1).

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, 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: A137966 A005498 A002222 * A001553 A009247 A093774

Adjacent sequences:  A006356 A006357 A006358 * A006360 A006361 A006362

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

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 July 26 08:24 EDT 2017. Contains 289799 sequences.