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%I #25 Jul 11 2023 12:38:18
%S 1,11,66,506,3641,26818,196119,1437799,10532302,77173602,565424068,
%T 4142793511,30353430420,222394369223,1629443428021,11938642758854,
%U 87472304803355,640893994357062,4695716053827835,34404674660198306
%N Number of distributive lattices; also number of paths with n turns when light is reflected from 11 glass plates.
%C Let M(11) be the 11 X 11 matrix (0,0,0,1)/(0,0,1,1)/(0,1,1,1)/(1,1,1,1) and let v(11) be the vector (1,1,1,1,1,1,1,1,1); then v(11)*M(11)^n = (x,y,z,t,u,v, w,m,n,o,a(n)) - _Benoit Cloitre_, Sep 29 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="/A030115/b030115.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://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 <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (6,15,-35,-35,56,28,-36,-9,10,1,-1).
%F G.f.: -(x -1)*(x^3 -x^2 -2*x +1)*(x^6 +x^5 -6*x^4 -6*x^3 +8*x^2 +8*x +1)/(x^11 -x^10 -10*x^9 +9*x^8 +36*x^7 -28*x^6 -56*x^5 +35*x^4 +35*x^3 -15*x^2 -6*x +1). [_Colin Barker_, Nov 09 2012]
%t CoefficientList[Series[-(x - 1) (x^3 - x^2 - 2 x + 1) (x^6 + x^5 - 6 x^4 - 6 x^3 + 8 x^2 + 8 x + 1)/(x^11 -x^10 - 10 x^9 + 9 x^8 + 36 x^7 - 28 x^6 - 56 x^5 + 35 x^4 + 35 x^3 - 15 x^2 - 6 x + 1), {x, 0, 30}], x] (* _Vincenzo Librandi_, Oct 19 2013 *)
%o (PARI) k=11; 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)
%Y See also A006356-A006359, A025030, A030112-A030116.
%K nonn,easy
%O 0,2
%A Jacques Haubrich (jhaubrich(AT)freeler.nl)
%E More terms from _Benoit Cloitre_, Sep 29 2002
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