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%I M4148 #42 Nov 07 2017 18:17:02
%S 1,6,21,91,371,1547,6405,26585,110254,457379,1897214,7869927,32645269,
%T 135416457,561722840,2330091144,9665485440,40093544735,166312629795,
%U 689883899612,2861717685450,11870733787751,49241167758705,204258021937291,847285745315256
%N Number of distributive lattices; also number of paths with n turns when light is reflected from 6 glass plates.
%C 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_, Apr 02 2002
%D J. Berman and P. Koehler, Cardinalities of finite distributive lattices, Mitteilungen aus dem Mathematischen Seminar Giessen, 121 (1976), 103-124.
%D 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 J. Haubrich, Multinacci Rijen [Multinacci sequences], Euclides (Netherlands), Vol. 74, Issue 4, 1998, pp. 131-133.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A006359/b006359.txt">Table of n, a(n) for n = 0..200</a>
%H J. Berman and P. Koehler, <a href="/A006356/a006356.pdf">Cardinalities of finite distributive lattices</a>, Mitteilungen aus dem Mathematischen Seminar Giessen, 121 (1976), 103-124. [Annotated scanned copy]
%H Emma L. L. Gao, Sergey Kitaev, Philip B. Zhang, <a href="https://personal.cis.strath.ac.uk/sergey.kitaev/index_files/Papers/pat-av-alt-words.pdf">Pattern-avoiding alternating words</a>, preprint, 2015.
%H G. Kreweras, <a href="http://www.numdam.org/item?id=MSH_1976__53__5_0">Les préordres totaux compatibles avec un ordre partiel</a>, Math. Sci. Humaines No. 53 (1976), 5-30.
%H G. Kreweras, <a href="/A019538/a019538.pdf">Les préordres totaux compatibles avec un ordre partiel</a>, Math. Sci. Humaines No. 53 (1976), 5-30. (Annotated scanned copy)
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (3,6,-4,-5,1,1).
%F G.f.: -(z^4 + z^3 - 3z^2 - 2z + 1) / (-1 + 3z + 6z^2 - 4z^3 - 5z^4 + z^5 + z^6). - M. Goebel (manfredg(AT)ICSI.Berkeley.EDU) Jul 26 1997
%F 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 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_, Oct 16 2002
%F 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_, Feb 06 2006
%F G.f.: 1/(-x-1/(-x-1/(-x-1/(-x-1/(-x-1/(-x-1)))))). - _Paul Barry_, Mar 24 2010
%p 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_, Mar 09 2007
%t LinearRecurrence[{3,6,-4,-5,1,1},{1,6,21,91,371,1547},30] (* _Harvey P. Dale_, Sep 03 2016 *)
%o (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 (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_, Feb 06 2006
%Y Cf. A000217, A000330, A050446, A050447, A006356, A006357, A006358.
%Y See also A025030, A030112-A030116.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
%E Alternative description from Jacques Haubrich (jhaubrich(AT)freeler.nl)
%E More terms from _James A. Sellers_, Dec 24 1999
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