A006358 Number of distributive lattices; also number of paths with n turns when light is reflected from 5 glass plates. (Formerly M3862)

1, 5, 15, 55, 190, 671, 2353, 8272, 29056, 102091, 358671, 1260143, 4427294, 15554592, 54648506, 191998646, 674555937, 2369942427, 8326406594, 29253473175, 102777312308, 361091343583, 1268635610806, 4457144547354

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.

D. E. Knuth, Art of Computer Programming, Vol. 3, Sect. 5.4.3, Column T1.

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 (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,changed

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