%I A006359 M4148
%S A006359 1,6,21,91,371,1547,6405,26585,110254,457379,1897214,7869927,32645269,
%T A006359 135416457,561722840,2330091144,9665485440,40093544735,166312629795,
%U A006359 689883899612,2861717685450,11870733787751,49241167758705
%N A006359 Number of distributive lattices; also number of paths with n turns when 
               light is reflected from 6 glass plates.
%C A006359 Let M denotes the 6 X 6 matrix = row by row (1,1,1,1,1,1)(1,1,1,1,1,0)(1,
               1,1,1,0,0)(1,1,1,0,0,0)(1,1,0,0,0,0)(1,0,0,0,0,0) and A(n) the vector 
               (x(n),y(n),z(n),t(n),u(n),v(n))=M^n*A where A is the vector (1,1,
               1,1,1,1) then a(n)=x(n) - Benoit Cloitre (benoit7848c(AT)orange.fr), 
               Apr 02 2002
%D A006359 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A006359 J. Berman and P. Koehler, Cardinalities of finite distributive lattices, 
               Mitteilungen aus dem Mathematischen Seminar Giessen, 121 (1976), 
               103-124.
%D A006359 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.
%D A006359 J. Haubrich, Multinacci Rijen [Multinacci sequences], Euclides (Netherlands), 
               Vol. 74, Issue 4, 1998, pp. 131-133.
%D A006359 G. Kreweras, Les preordres totaux compatibles avec un ordre partiel. 
               Math. Sci. Humaines No. 53 (1976), 5-30.
%F A006359 G.f. from M. Goebel (manfredg(AT)ICSI.Berkeley.EDU) Jul 26 1997: -(z^4 
               + z^3 - 3z^2 - 2z + 1) / (-1 + 3z + 6z^2 - 4z^3 - 5z^4 + z^5 + z^6).
%F A006359 a(n)=3*a(n-1)+6*a(n-2)-4*a(n-3)-5*a(n-4)+a(n-5)+a(n-6).
%F A006359 a(n) is asymptotic to z(6)*w(6)^n where w(6)=(1/2)/cos(6*Pi/13) and z(6) 
               is the root 1<x<2 of P(6, X) = -1-91*X+2366*X^2+26364*X^3-142805*X^4-371293*X^5+371293*X^6 
               - Benoit Cloitre (benoit7848c(AT)orange.fr), Oct 16 2002
%F A006359 G.f.: A(x) = (1+3*x-3*x^2-4*x^3+x^4+x^5)/(1-3*x-6*x^2+4*x^3+5*x^4-x^5-x^6). 
               - Paul D. Hanna (pauldhanna(AT)juno.com), Feb 06 2006
%p A006359 A=seq(a.j,j=0..5):grammar1:=[Q5,{ seq(Q.i=Union(Epsilon,seq(Prod(a.j,
               Q.j),j=5-i..5)),i=0..5), seq(a.j=Z,j=0..5) }, unlabeled]: seq(count(grammar1,
               size=j),j=0..22); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Mar 09 2007
%o A006359 (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)
%o A006359 (PARI) {a(n)=local(p=6);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 (pauldhanna(AT)juno.com), Feb 06 2006
%Y A006359 Cf. A000217, A000330, A050446, A050447, A006356, A006357, A006358.
%Y A006359 See also A025030, A030112-A030116.
%Y A006359 Sequence in context: A137966 A005498 A002222 this_sequence A001553 A009247 
               A093774
%Y A006359 Adjacent sequences: A006356 A006357 A006358 this_sequence A006360 A006361 
               A006362
%K A006359 nonn,easy
%O A006359 0,2
%A A006359 N. J. A. Sloane (njas(AT)research.att.com).
%E A006359 Alternative description from Jacques Haubrich (jhaubrich(AT)freeler.nl).
%E A006359 More terms from James A. Sellers (sellersj(AT)math.psu.edu), Dec 24 1999