A030115 Number of distributive lattices; also number of paths with n turns when light is reflected from 11 glass plates.

1, 11, 66, 506, 3641, 26818, 196119, 1437799, 10532302, 77173602, 565424068, 4142793511, 30353430420, 222394369223, 1629443428021, 11938642758854, 87472304803355, 640893994357062, 4695716053827835, 34404674660198306

Offset: 0

Author

Jacques Haubrich (jhaubrich(AT)freeler.nl)

Keywords

• nonn
• easy

Comments

Let M(11) be the 11 X 11 matrix (0,0,0,1)/(0,0,1,1)/(0,1,1,1)/(1,1,1,1) and let v(11) be the vector (1,1,1,1,1,1,1,1,1); then v(11)*M(11)^n = (x,y,z,t,u,v, w,m,n,o,a(n)) - Benoit Cloitre, Sep 29 2002

References

• J. Berman and P. Koehler, Cardinalities of finite distributive lattices, Mitteilungen aus dem Mathematischen Seminar Giessen, 121 (1976), 103-124.
• J. Haubrich, Multinacci Rijen [Multinacci sequences], Euclides (Netherlands), Vol. 74, Issue 4, 1998, pp. 131-133.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• J. Berman and P. Koehler, Cardinalities of finite distributive lattices, Mitteilungen aus dem Mathematischen Seminar Giessen, 121 (1976), 103-124. [Annotated scanned copy]
• G. Kreweras, Les préordres totaux compatibles avec un ordre partiel, Math. Sci. Humaines No. 53 (1976), 5-30.
• Index entries for linear recurrences with constant coefficients, signature (6,15,-35,-35,56,28,-36,-9,10,1,-1).

Crossrefs

See also A006356-A006359, A025030, A030112-A030116.

Programs

Mathematica

CoefficientList[Series[-(x - 1) (x^3 - x^2 - 2 x + 1) (x^6 + x^5 - 6 x^4 - 6 x^3 + 8 x^2 + 8 x + 1)/(x^11 -x^10 - 10 x^9 + 9 x^8 + 36 x^7 - 28 x^6 - 56 x^5 + 35 x^4 + 35 x^3 - 15 x^2 - 6 x + 1), {x, 0, 30}], x] (* Vincenzo Librandi, Oct 19 2013 *)

PARI

k=11; 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)

Formula

G.f.: -(x -1)*(x^3 -x^2 -2*x +1)*(x^6 +x^5 -6*x^4 -6*x^3 +8*x^2 +8*x +1)/(x^11 -x^10 -10*x^9 +9*x^8 +36*x^7 -28*x^6 -56*x^5 +35*x^4 +35*x^3 -15*x^2 -6*x +1). [Colin Barker, Nov 09 2012]

Extensions

More terms from Benoit Cloitre, Sep 29 2002