A025030 Number of distributive lattices; also number of paths with n turns when light is reflected from 7 glass plates.

1, 7, 28, 140, 658, 3164, 15106, 72302, 345775, 1654092, 7911970, 37846314, 181033035, 865951710, 4142180085, 19813648817, 94776329265, 453351783116, 2168556616440, 10373043626906, 49618272850056, 237343357526002

Offset: 0

Author

Jacques Haubrich (jhaubrich(AT)freeler.nl)

Keywords

• nonn
• easy

Comments

Let M(7) be the 7 X 7 matrix: (0,0,0,0,0,0,1)/(0,0,0,0,0,1,1)/(0,0,0,0,1,1,1)/(0,0,0,1,1,1,1)/(0,0,1,1,1,1,1)/(0,1,1,1,1,1,1)/(1,1,1,1,1,1,1) and let v(7) be the vector (1,1,1,1,1,1,1); then v(7)*M(7)^n = (x,y,z,t,u,v,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 (4,6,-10,-5,6,1,-1).

Crossrefs

See also A006356-A006359, A030112-A030116.

Programs

Magma

I:=[1, 7, 28, 140, 658, 3164, 15106]; [n le 7 select I[n] else 4*Self(n-1)+6*Self(n-2)-10*Self(n-3)-5*Self(n-4)+6*Self(n-5)+Self(n-6)-Self(n-7): n in [1..30]]; // Vincenzo Librandi, Apr 22 2012

Mathematica

CoefficientList[Series[(1+3*x-6*x^2-4*x^3+5*x^4+x^5-x^6)/((1-x)*(1+x-x^2)*(1-4*x-4*x^2+x^3+x^4)),{x,0,30}],x] (* Vincenzo Librandi, Apr 22 2012 *)
LinearRecurrence[{4,6,-10,-5,6,1,-1},{1,7,28,140,658,3164,15106},30] (* Harvey P. Dale, Feb 26 2023 *)

PARI

k=7; 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) + 6*a(n-2) - 10*a(n-3) - 5*a(n-4) + 6*a(n-5) + a(n-6) - a(n-7).

a(n) is asymptotic to z(7)*w(7)^n where w(7) = (1/2)/cos(7*Pi/15) and z(7) is the root 1 < x < 2 of P(7, X) = 1 - 120*X - 8100*X^2 - 57375*X^3 + 50625*X^4. - Benoit Cloitre, Oct 16 2002

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

Extensions

More terms from Benoit Cloitre, Sep 29 2002