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

0,2

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).

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

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 *)

Prog

(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)
(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

Crossrefs

See also A006356-A006359, A030112-A030116.

Sequence in context: A249872 A238448 A290356 * A001554 A370243 A370103

Adjacent sequences: A025027 A025028 A025029 * A025031 A025032 A025033

Keyword

nonn,easy

Author

Jacques Haubrich (jhaubrich(AT)freeler.nl)

Extensions

More terms from Benoit Cloitre, Sep 29 2002

Status

approved