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A006358 Number of distributive lattices; also number of paths with n turns when light is reflected from 5 glass plates.
(Formerly M3862)
10
1, 5, 15, 55, 190, 671, 2353, 8272, 29056, 102091, 358671, 1260143, 4427294, 15554592, 54648506, 191998646, 674555937, 2369942427, 8326406594, 29253473175, 102777312308, 361091343583, 1268635610806, 4457144547354 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Let M denotes the 5 X 5 matrix = row by row (1,1,1,1,1)(1,1,1,1,0)(1,1,1,0,0)(1,1,0,0,0)(1,0,0,0,0) and A(n) the vector (x(n),y(n),z(n),t(n),u(n))=M^n*A where A is the vector (1,1,1,1,1); then a(n)=y(n). - Benoit Cloitre, Apr 02 2002

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, Philip B. Zhang, Pattern-avoiding alternating words, preprint, 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 preordres totaux compatibles avec un ordre partiel, Math. Sci. Humaines No. 53 (1976), 5-30.

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 1992.

Index entries for linear recurrences with constant coefficients, signature (3,3,-4,-1,1).

FORMULA

a(n) = 3*a(n-1) + 3*a(n-2) - 4*a(n-3) - a(n-4) + a(n-5).

a(n) is asymptotic to z(5)*w(5)^n where w(5)=(1/2)/cos(5*Pi/11) and z(5) is the root 1<x<2 of P(5, X) = -1 + 55*X + 847*X^2 - 5324*X^3 - 14641*X^4 + 14641*X^5. - Benoit Cloitre, Oct 16 2002

G.f.: A(x) = (1 + 2*x - 3*x^2 - x^3 + x^4)/(1 - 3*x - 3*x^2 + 4*x^3 + x^4 - x^5). - Paul D. Hanna, Feb 06 2006

MAPLE

A=seq(a.j, j=0..4):grammar1:=[Q4, { seq(Q.i=Union(Epsilon, seq(Prod(a.j, Q.j), j=4-i..4)), i=0..4), seq(a.j=Z, j=0..4) }, unlabeled]: seq(count(grammar1, size=j), j=0..23); # Zerinvary Lajos, Mar 09 2007

A006358:=-(z-1)*(z**3-3*z-1)/(-1+3*z+3*z**2-4*z**3-z**4+z**5); # conjectured by Simon Plouffe in his 1992 dissertation

MATHEMATICA

m = Table[ If[j <= 6-i, 1, 0], {i, 1, 5}, {j, 1, 5}] ; a[n_] := MatrixPower[m, n].Table[1, {5}]; Table[ a[n], {n, 0, 23}][[All, 1]] (* Jean-François Alcover, Dec 08 2011, after Benoit Cloitre *)

LinearRecurrence[{3, 3, -4, -1, 1}, {1, 5, 15, 55, 190}, 30] (* Harvey P. Dale, Jun 16 2016 *)

PROG

(PARI) k=5; 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)

(PARI) {a(n)=local(p=5); 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)}

CROSSREFS

Cf. A000217, A000330, A050446, A050447.

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

Cf. A038201 (5-wave sequence).

Sequence in context: A002221 A007714 A123011 * A054108 A149585 A114947

Adjacent sequences:  A006355 A006356 A006357 * A006359 A006360 A006361

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

Alternative description and formula from Jacques Haubrich (jhaubrich(AT)freeler.nl)

More terms from James A. Sellers, Dec 24 1999

STATUS

approved

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