OFFSET
0,2
COMMENTS
We do not assume all maximal elements have maximal rank and thus use graded poset to mean: For every element x, all maximal chains among those with x as greatest element have the same finite length.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Vladimir Retakh, Shirlei Serconek, and Robert Wilson, Hilbert series of algebras associated to direct graphs and order homology, arXiv 1010.6295 [math.RA], 2010-2011.
Wikipedia, Graded poset
Index entries for linear recurrences with constant coefficients, signature (7,-10,3).
FORMULA
a(n+3) = 7a(n+2) - 10a(n+1) + 3a(n), a(0)=1, a(1)=2, a(2)=8.
G.f.: (1-5x+4x^2)/(1-7x+10x^2-3x^3).
MATHEMATICA
m = {{3, 3, 1, 0}, {1, 3, 0, 0}, {2, 3, 1, 0}, {6, 9, 2, 0}}; Table[MatrixPower[m, n][[4, 3]], {n, 1, 40}]
PROG
(Python)
def a(n, adict={0:1, 1:2, 2:8}):
if n in adict:
return adict[n]
adict[n]=7*a(n-1)-10*a(n-2)+3*a(n-3)
return adict[n]
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
David Nacin, Feb 13 2012
STATUS
approved