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%I #39 Sep 08 2022 08:44:50
%S 1,8,36,204,1086,5916,31998,173502,940005,5094220,27604798,149590922,
%T 810627389,4392774126,23804329059,128995094597,699021261776,
%U 3787979292364,20526967746120,111235140046330,602780523265720,3266453022809170,17700829632401740,95920366069513405
%N Number of distributive lattices; also number of paths with n turns when light is reflected from 8 glass plates.
%C Let M(8) be the 8 X 8 matrix (0,0,0,0,0,0,0,1)/(0,0,0,0,0,0,1,1)/(0,0,0,0,0,1,1,1)/(0,0,0,0,1,1,1,1)/(0,0,0,1,1,1,1,1)/(0,0,1,1,1,1,1,1)/(0,1,1,1,1,1,1,1)/(1,1,1,1,1,1,1,1) and let v(8) be the vector (1,1,1,1,1,1,1,1); then v(8)*M(8)^n = (x,y,z,t,u,v, w,a(n)). - _Benoit Cloitre_, Sep 29 2002
%C For a k-glass sequence, say a(n,k), a(n,k) is always asymptotic to z(k)*w(k)^n where w(k)=(1/2)/cos(k*Pi/(2k+1)) and it is conjectured that z(k) is the root 1<x<2 of a polynomial of degree Phi(2k+1)/2. - _Benoit Cloitre_, Oct 16 2002
%D J. Berman and P. Koehler, Cardinalities of finite distributive lattices, Mitteilungen aus dem Mathematischen Seminar Giessen, 121 (1976), 103-124.
%D J. Haubrich, Multinacci Rijen [Multinacci sequences], Euclides (Netherlands), Vol. 74, Issue 4, 1998, pp. 131-133.
%H Vincenzo Librandi, <a href="/A030112/b030112.txt">Table of n, a(n) for n = 0..1000</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 G. Kreweras, <a href="http://archive.numdam.org/article/MSH_1976__53__5_0.pdf">Les préordres totaux compatibles avec un ordre partiel</a>, Math. Sci. Humaines No. 53 (1976), 5-30.
%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (4,10,-10,-15,6,7,-1,-1).
%F a(n) = 4*a(n-1)+ 10*a(n-2)-10*a(n-3)-15*a(n-4)+ 6*a(n-5)+7*a(n-6)-a(n-7)-a(n-8). - _Benoit Cloitre_, Oct 09 2002
%F a(n) is asymptotic to z(8)*w(8)^n where w(8)=(1/2)/cos(8*Pi/17) and z(8) is the root 1<x<2 of P(8, X) = 1 +204*X -12138*X^2 -324258*X^3 +4593655*X^4+36916282X^5 -168962983*X^6 -410338673*X^7 +410338673*X^8. - _Benoit Cloitre_, Oct 16 2002
%F G.f.: (1+x)*(1-x-x^2)*(1+4*x-4*x^2-x^3+x^4)/(1-4*x-10*x^2+10*x^3+15*x^4-6*x^5-7*x^6+x^7+x^8). - _Colin Barker_, Mar 31 2012
%p nmax:=20: with(LinearAlgebra): M:=Matrix([[0,0,0,0,0,0,0,1], [0,0,0,0,0,0,1,1], [0,0,0,0,0,1,1,1], [0,0,0,0,1,1,1,1], [0,0,0,1,1,1,1,1], [0,0,1,1,1,1,1,1], [0,1,1,1,1,1,1,1], [1,1,1,1,1,1,1,1]]): v:= Vector[row]([1,1,1,1,1,1,1,1]): for n from 0 to nmax do b:=evalm(v&*M^n): a(n):=b[8] od: seq(a(n), n=0..nmax); # _Johannes W. Meijer_, Aug 03 2011
%t CoefficientList[Series[(1+x)*(1-x-x^2)*(1+4*x-4*x^2-x^3+x^4)/(1-4*x-10*x^2+10*x^3+15*x^4-6*x^5-7*x^6+x^7+x^8),{x,0,30}],x] (* _Vincenzo Librandi_, Apr 22 2012 *)
%o (PARI) k=8; 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 (Magma) I:=[1, 8, 36, 204, 1086, 5916, 31998, 173502]; [n le 8 select I[n] else 4*Self(n-1)+10*Self(n-2)-10*Self(n-3)-15*Self(n-4)+6*Self(n-5)+7*Self(n-6)-Self(n-7)-Self(n-8): n in [1..25]]; // _Vincenzo Librandi_, Apr 22 2012
%Y See also A006356-A006359, A025030, A030113-A030116.
%K nonn,easy
%O 0,2
%A Jacques Haubrich (jhaubrich(AT)freeler.nl)
%E More terms from _Benoit Cloitre_, Sep 29 2002
%E Comment corrected by _Johannes W. Meijer_, Aug 03 2011
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