OFFSET
0,2
COMMENTS
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
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
REFERENCES
J. Berman and P. Koehler, Cardinalities of finite distributive lattices, Mitteilungen aus dem Mathematischen Seminar Giessen, 121 (1976), 103-124.
J. Haubrich, Multinacci Rijen [Multinacci sequences], Euclides (Netherlands), Vol. 74, Issue 4, 1998, pp. 131-133.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
J. Berman and P. Koehler, Cardinalities of finite distributive lattices, Mitteilungen aus dem Mathematischen Seminar Giessen, 121 (1976), 103-124. [Annotated scanned copy]
G. Kreweras, Les préordres totaux compatibles avec un ordre partiel, Math. Sci. Humaines No. 53 (1976), 5-30.
Index entries for linear recurrences with constant coefficients, signature (4,10,-10,-15,6,7,-1,-1).
FORMULA
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
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
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
MAPLE
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
MATHEMATICA
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 *)
PROG
(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)
(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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Jacques Haubrich (jhaubrich(AT)freeler.nl)
EXTENSIONS
More terms from Benoit Cloitre, Sep 29 2002
Comment corrected by Johannes W. Meijer, Aug 03 2011
STATUS
approved