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A030112 Number of distributive lattices; also number of paths with n turns when light is reflected from 8 glass plates. 11
1, 8, 36, 204, 1086, 5916, 31998, 173502, 940005, 5094220, 27604798, 149590922, 810627389, 4392774126, 23804329059, 128995094597, 699021261776, 3787979292364, 20526967746120, 111235140046330, 602780523265720 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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 1<x<2 of a polynomial of degree Phi(2k+1)/2. - Benoit 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 preordres 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).

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 1<x<2 of P(8, X) = 1 +204*X -12138*X^2 -324258*X^3 +4593655*X^4+36916282X^5 -168962983*X^6 -410338673*X^7 +410338673*X^8. - Benoit 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

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 *)

PROG

(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)

(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

CROSSREFS

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

Sequence in context: A079819 A238815 A290357 * A001555 A032770 A032794

Adjacent sequences:  A030109 A030110 A030111 * A030113 A030114 A030115

KEYWORD

nonn,easy

AUTHOR

Jacques Haubrich (jhaubrich(AT)freeler.nl)

EXTENSIONS

More terms from Benoit Cloitre, Sep 29 2002

Comment corrected by Johannes W. Meijer, Aug 03 2011

STATUS

approved

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Last modified August 18 14:09 EDT 2017. Contains 290720 sequences.