A030112 Number of distributive lattices; also number of paths with n turns when light is reflected from 8 glass plates.

1, 8, 36, 204, 1086, 5916, 31998, 173502, 940005, 5094220, 27604798, 149590922, 810627389, 4392774126, 23804329059, 128995094597, 699021261776, 3787979292364, 20526967746120, 111235140046330, 602780523265720, 3266453022809170, 17700829632401740, 95920366069513405

Offset: 0

Author

Jacques Haubrich (jhaubrich(AT)freeler.nl)

Keywords

• nonn
• easy

Comments

Let M(8) be the 8 X 8 matrix (0,0,0,0,0,0,0,1)/(0,0,0,0,0,0,1,1)/(0,0,0,0,0,1,1,1)/(0,0,0,0,1,1,1,1)/(0,0,0,1,1,1,1,1)/(0,0,1,1,1,1,1,1)/(0,1,1,1,1,1,1,1)/(1,1,1,1,1,1,1,1) and let v(8) be the vector (1,1,1,1,1,1,1,1); then v(8)*M(8)^n = (x,y,z,t,u,v, w,a(n)). - Benoit Cloitre, Sep 29 2002

For a k-glass sequence, say a(n,k), a(n,k) is always asymptotic to z(k)*w(k)^n where w(k)=(1/2)/cos(k*Pi/(2k+1)) and it is conjectured that z(k) is the root 1Benoit Cloitre, Oct 16 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 (4,10,-10,-15,6,7,-1,-1).

Crossrefs

See also A006356-A006359, A025030, A030113-A030116.

Programs

Magma

I:=[1, 8, 36, 204, 1086, 5916, 31998, 173502]; [n le 8 select I[n] else 4*Self(n-1)+10*Self(n-2)-10*Self(n-3)-15*Self(n-4)+6*Self(n-5)+7*Self(n-6)-Self(n-7)-Self(n-8):  n in [1..25]]; // Vincenzo Librandi, Apr 22 2012

Maple

nmax:=20: with(LinearAlgebra): M:=Matrix([[0,0,0,0,0,0,0,1], [0,0,0,0,0,0,1,1], [0,0,0,0,0,1,1,1], [0,0,0,0,1,1,1,1], [0,0,0,1,1,1,1,1], [0,0,1,1,1,1,1,1], [0,1,1,1,1,1,1,1], [1,1,1,1,1,1,1,1]]): v:= Vector[row]([1,1,1,1,1,1,1,1]): for n from 0 to nmax do b:=evalm(v&*M^n): a(n):=b[8] od: seq(a(n), n=0..nmax); # Johannes W. Meijer, Aug 03 2011

Mathematica

CoefficientList[Series[(1+x)*(1-x-x^2)*(1+4*x-4*x^2-x^3+x^4)/(1-4*x-10*x^2+10*x^3+15*x^4-6*x^5-7*x^6+x^7+x^8),{x,0,30}],x] (* Vincenzo Librandi, Apr 22 2012 *)

PARI

k=8; 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

a(n) = 4*a(n-1)+ 10*a(n-2)-10*a(n-3)-15*a(n-4)+ 6*a(n-5)+7*a(n-6)-a(n-7)-a(n-8). - Benoit Cloitre, Oct 09 2002

a(n) is asymptotic to z(8)*w(8)^n where w(8)=(1/2)/cos(8*Pi/17) and z(8) is the root 1Benoit Cloitre, Oct 16 2002

G.f.: (1+x)*(1-x-x^2)*(1+4*x-4*x^2-x^3+x^4)/(1-4*x-10*x^2+10*x^3+15*x^4-6*x^5-7*x^6+x^7+x^8). - Colin Barker, Mar 31 2012

Extensions

More terms from Benoit Cloitre, Sep 29 2002

Comment corrected by Johannes W. Meijer, Aug 03 2011