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Number of distributive lattices; also number of paths with n turns when light is reflected from 9 glass plates.
3

%I #26 Jul 11 2023 12:38:01

%S 1,9,45,285,1695,10317,62349,377739,2286648,13846117,83833256,

%T 507596153,3073376281,18608642427,112671254094,682200039446,

%U 4130572919575,25009722123505,151428434581516,916866281219258

%N Number of distributive lattices; also number of paths with n turns when light is reflected from 9 glass plates.

%C Let M(9) be the 9 X 9 matrix (0,0,0,1)/(0,0,1,1)/(0,0,1,1)/(1,1,1,1) and let v(9) be the vector (1,1,1,1,1,1,1,1,1); then v(9)*M(9)^n = (x,y,z,t,u,v, w,m,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="/A030113/b030113.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_09">Index entries for linear recurrences with constant coefficients</a>, signature (5,10,-20,-15,21,7,-8,-1,1).

%F G.f.: -(x^8 -x^7 -7*x^6 +6*x^5 +15*x^4 -10*x^3 -10*x^2 +4*x +1)/(x^9 -x^8 -8*x^7 +7*x^6 +21*x^5 -15*x^4 -20*x^3 +10*x^2 +5*x -1). [_Colin Barker_, Nov 09 2012]

%t CoefficientList[Series[-(x^8 - x^7 -7 x^6 + 6 x^5 + 15 x^4 - 10 x^3 - 10 x^2 + 4 x + 1)/(x^9 - x^8 - 8 x^7 + 7 x^6 + 21 x^5 - 15 x^4 - 20 x^3 + 10 x^2 + 5 x - 1), {x, 0, 30}], x] (* _Vincenzo Librandi_, Oct 19 2013 *)

%t LinearRecurrence[{5,10,-20,-15,21,7,-8,-1,1},{1,9,45,285,1695,10317,62349,377739,2286648},30] (* _Harvey P. Dale_, Dec 13 2015 *)

%o (PARI) k=9; 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