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Number of distributive lattices; also number of paths with n turns when light is reflected from 7 glass plates.
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%I #34 Feb 26 2023 12:56:14

%S 1,7,28,140,658,3164,15106,72302,345775,1654092,7911970,37846314,

%T 181033035,865951710,4142180085,19813648817,94776329265,453351783116,

%U 2168556616440,10373043626906,49618272850056,237343357526002

%N Number of distributive lattices; also number of paths with n turns when light is reflected from 7 glass plates.

%C Let M(7) be the 7 X 7 matrix: (0,0,0,0,0,0,1)/(0,0,0,0,0,1,1)/(0,0,0,0,1,1,1)/(0,0,0,1,1,1,1)/(0,0,1,1,1,1,1)/(0,1,1,1,1,1,1)/(1,1,1,1,1,1,1) and let v(7) be the vector (1,1,1,1,1,1,1); then v(7)*M(7)^n = (x,y,z,t,u,v,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="/A025030/b025030.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_07">Index entries for linear recurrences with constant coefficients</a>, signature (4,6,-10,-5,6,1,-1).

%F a(n) = 4*a(n-1) + 6*a(n-2) - 10*a(n-3) - 5*a(n-4) + 6*a(n-5) + a(n-6) - a(n-7).

%F a(n) is asymptotic to z(7)*w(7)^n where w(7) = (1/2)/cos(7*Pi/15) and z(7) is the root 1 < x < 2 of P(7, X) = 1 - 120*X - 8100*X^2 - 57375*X^3 + 50625*X^4. - _Benoit Cloitre_, Oct 16 2002

%F G.f.: (1 + 3*x - 6*x^2 - 4*x^3 + 5*x^4 + x^5 - x^6)/((1 - x)*(1 + x - x^2)*(1 - 4*x - 4*x^2 + x^3 + x^4)). - _Colin Barker_, Mar 31 2012

%t CoefficientList[Series[(1+3*x-6*x^2-4*x^3+5*x^4+x^5-x^6)/((1-x)*(1+x-x^2)*(1-4*x-4*x^2+x^3+x^4)),{x,0,30}],x] (* _Vincenzo Librandi_, Apr 22 2012 *)

%t LinearRecurrence[{4,6,-10,-5,6,1,-1},{1,7,28,140,658,3164,15106},30] (* _Harvey P. Dale_, Feb 26 2023 *)

%o (PARI) k=7; 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, 7, 28, 140, 658, 3164, 15106]; [n le 7 select I[n] else 4*Self(n-1)+6*Self(n-2)-10*Self(n-3)-5*Self(n-4)+6*Self(n-5)+Self(n-6)-Self(n-7): n in [1..30]]; // _Vincenzo Librandi_, Apr 22 2012

%Y See also A006356-A006359, A030112-A030116.

%K nonn,easy

%O 0,2

%A Jacques Haubrich (jhaubrich(AT)freeler.nl)

%E More terms from _Benoit Cloitre_, Sep 29 2002