A006357 Number of distributive lattices; also number of paths with n turns when light is reflected from 4 glass plates. (Formerly M3396)

1, 4, 10, 30, 85, 246, 707, 2037, 5864, 16886, 48620, 139997, 403104, 1160693, 3342081, 9623140, 27708726, 79784098, 229729153, 661478734, 1904652103, 5484227157, 15791202736, 45468956106, 130922641160, 376976720745, 1085461206128, 3125460977225

Offset

0,2

Comments

Let M denotes the 4 X 4 matrix = row by row (1,1,1,1)(1,1,1,0)(1,1,0,0)(1,0,0,0) and A(n) the vector (x(n),y(n),z(n),t(n))=M^n*A where A is the vector (1,1,1,1) then a(n)=x(n). - Benoit Cloitre, Apr 02 2002

In general, the g.f. for p glass plates is A(x) = F_{p-1}(-x)/F_p(x) where F_p(x) = Sum_{k=0,p} (-1)^[(k+1)/2]*C([(p+k)/2],k)*x^k. - Paul D. Hanna, Feb 06 2006

a(n)/a(n-1) tends to 2.879385..., the longest diagonal of a nonagon with edge 1; or: sin(4*Pi/9)/sin(Pi/9). The sequence is the INVERT transform of (1, 3, 3, 5, 8, 13, 21, 34, 55, 89, 144, ...). - Gary W. Adamson, Jul 16 2015

References

J. Berman and P. Koehler, Cardinalities of finite distributive lattices, Mitteilungen aus dem Mathematischen Seminar Giessen, 121 (1976), 103-124.

S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (see p. 120).

J. Haubrich, Multinacci Rijen [Multinacci sequences], Euclides (Netherlands), Vol. 74, Issue 4, 1998, pp. 131-133.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

T. D. Noe, Table of n, a(n) for n = 0..200

J. Berman and P. Koehler, Cardinalities of finite distributive lattices, Mitteilungen aus dem Mathematischen Seminar Giessen, 121 (1976), 103-124. [Annotated scanned copy]

Emma L. L. Gao, Sergey Kitaev, and Philip B. Zhang, Pattern-avoiding alternating words, arXiv:1505.04078 [math.CO], 2015.

Manfred Goebel, Rewriting Techniques and Degree Bounds for Higher Order Symmetric Polynomials, Applicable Algebra in Engineering, Communication and Computing (AAECC), Volume 9, Issue 6 (1999), 559-573.

G. Kreweras, Les préordres totaux compatibles avec un ordre partiel, Math. Sci. Humaines No. 53 (1976), 5-30.

G. Kreweras, Les préordres totaux compatibles avec un ordre partiel, Math. Sci. Humaines No. 53 (1976), 5-30. (Annotated scanned copy)

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992

Index entries for linear recurrences with constant coefficients, signature (2,3,-1,-1).

Formula

G.f.: (1 + 2*x - x^2 - x^3)/( (1 +x)*(1 -3*x +x^3) ). - Simon Plouffe in his 1992 dissertation

a(n) = 2*a(n-1) + 3*a(n-2) - a(n-3) - a(n-4).

a(n) is asymptotic to z(4)*w(4)^n where w(4) = (1/2)/cos(4*Pi/9) and z(4) is the root 1 < x < 2 of P(4, X) = 1 + 27*X - 324*X^2 + 243*X^3. - Benoit Cloitre, Oct 16 2002

Binomial transform of A122167(unsigned): (1, 3, 3, 11, 10, 40, 33, 146, ...). - Gary W. Adamson, Nov 24 2007

G.f.: 1/(-x-1/(-x-1/(-x-1/(-x-1)))). - Paul Barry, Mar 24 2010

Mathematica

LinearRecurrence[{2, 3, -1, -1}, {1, 4, 10, 30}, 30] (* Harvey P. Dale, Nov 18 2013 *)

Prog

(PARI) a(n)=local(p=4); polcoeff(sum(k=0, p-1, (-1)^((k+1)\2)*binomial((p+k-1)\2, k)* (-x)^k)/sum(k=0, p, (-1)^((k+1)\2)*binomial((p+k)\2, k)*x^k+x*O(x^n)), n) \\ Paul D. Hanna

Crossrefs

Cf. A000217, A000330, A050446, A050447, A006356-A006359, A025030, A030112-A030116, A122167, A091024.

Cf. A038197 (4-wave sequence).

Sequence in context: A007713 A058488 A036674 * A114946 A243793 A001551

Adjacent sequences: A006354 A006355 A006356 * A006358 A006359 A006360

Keyword

nonn,nice,easy

Author

N. J. A. Sloane

Extensions

Recurrence, alternative description from Jacques Haubrich (jhaubrich(AT)freeler.nl)

More terms from James A. Sellers, Dec 24 1999

More terms from Paul D. Hanna, Feb 06 2006

Status

approved