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%I #18 Jul 11 2023 12:38:13
%S 1,12,78,650,5083,40690,323401,2576795,20514715,163369570,1300879372,
%T 10358963615,82488063476,656851828075,5230500095281,41650400765615,
%U 331661528811227,2641015991983270,21030372117368865,167464549591889570,1333517788817519126
%N Number of distributive lattices; also number of paths with n turns when light is reflected from 12 glass plates.
%C Let M(12) be the 12 X 12 matrix (0,0,0,1)/(0,0,1,1)/(0,1,1,1)/(1,1,1,1) and let v(12) be the 12-vector (1,1,..,1,1,1); then v(12)*M(12)^n = (x(1),x(2),...x(11),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 T. D. Noe, <a href="/A030116/b030116.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 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.
%F G.f.: 1/(-x-1/(-x-1/(-x-1/(-x-1/(-x-1/(-x-1/(-x-1/(-x-1/(-x-1/(-x- 1/(-x-1/(-x-1)))))))))))). [_Paul Barry_, Mar 24 2010]
%o (PARI) k=12; 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-A030115.
%K nonn
%O 0,2
%A Jacques Haubrich (jhaubrich(AT)freeler.nl)
%E More terms from _Benoit Cloitre_, Sep 29 2002
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