A030113 Number of distributive lattices; also number of paths with n turns when light is reflected from 9 glass plates.

1, 9, 45, 285, 1695, 10317, 62349, 377739, 2286648, 13846117, 83833256, 507596153, 3073376281, 18608642427, 112671254094, 682200039446, 4130572919575, 25009722123505, 151428434581516, 916866281219258

Offset: 0

Author

Jacques Haubrich (jhaubrich(AT)freeler.nl)

Keywords

• nonn
• easy

Comments

Let M(9) be the 9 X 9 matrix (0,0,0,1)/(0,0,1,1)/(0,0,1,1)/(1,1,1,1) and let v(9) be the vector (1,1,1,1,1,1,1,1,1); then v(9)*M(9)^n = (x,y,z,t,u,v, w,m,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 (5,10,-20,-15,21,7,-8,-1,1).

Crossrefs

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

Programs

Mathematica

CoefficientList[Series[-(x^8 - x^7 -7 x^6 + 6 x^5 + 15 x^4 - 10 x^3 - 10 x^2 + 4 x + 1)/(x^9 - x^8 - 8 x^7 + 7 x^6 + 21 x^5 - 15 x^4 - 20 x^3 + 10 x^2 + 5 x - 1), {x, 0, 30}], x] (* Vincenzo Librandi, Oct 19 2013 *)
LinearRecurrence[{5,10,-20,-15,21,7,-8,-1,1},{1,9,45,285,1695,10317,62349,377739,2286648},30] (* Harvey P. Dale, Dec 13 2015 *)

PARI

k=9; 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^8 -x^7 -7*x^6 +6*x^5 +15*x^4 -10*x^3 -10*x^2 +4*x +1)/(x^9 -x^8 -8*x^7 +7*x^6 +21*x^5 -15*x^4 -20*x^3 +10*x^2 +5*x -1). [Colin Barker, Nov 09 2012]

Extensions

More terms from Benoit Cloitre, Sep 29 2002