A006359 Number of distributive lattices; also number of paths with n turns when light is reflected from 6 glass plates.

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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.
• 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, and Philip B. Zhang, Pattern-avoiding alternating words, arXiv:1505.04078 [math.CO], 2015.
• 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)
• Index entries for linear recurrences with constant coefficients, signature (3,6,-4,-5,1,1).

Crossrefs

Cf. A000217, A000330, A050446, A050447, A006356, A006357, A006358.

See also A025030, A030112, A030113, A030114, A030115, A030116.

Programs

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 *)

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

Formula

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

Extensions

Alternative description from Jacques Haubrich (jhaubrich(AT)freeler.nl)

More terms from James Sellers, Dec 24 1999