OFFSET
0,3
COMMENTS
In general, a(n) = (2/m)*Sum_{k=1..m-1} sin(Pi*k/m)^2(1+2*cos(Pi*k/m))^n counts the (s(0), s(1), ..., s(n)) such that 0 < s(i) < m and |s(i) - s(i-1)| <= 1 for i = 1,2,...,n, s(0) = 1, s(n) = 1. Here is m=8.
a(n) is the number of Motzkin n-paths of height <= 6. - Alois P. Heinz, Nov 24 2023
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..2203
S. Felsner and D. Heldt, Lattice Path Enumeration and Toeplitz Matrices, J. Int. Seq. 18 (2015) # 15.1.3.
Daniel Heldt, On the mixing time of the face flip-and up/down Markov chain for some families of graphs, Dissertation, Mathematik und Naturwissenschaften der Technischen Universität Berlin zur Erlangung des akademischen Grades Doktor der Naturwissenschaften, 2016.
FORMULA
a(n) = (1/4)*Sum_{k=1..7} sin(Pi*k/8)^2*(1+2*cos(Pi*k/8))^n.
Conjecture: a(n)= +7*a(n-1) -15*a(n-2) +5*a(n-3) +15*a(n-4) -9*a(n-5) -3*a(n-6) +a(n-7) with g.f. -x*(1-5*x+5*x^2+6*x^3-7*x^4-2*x^5+x^6) / ( (x-1)*(x^2+2*x-1)*(x^4-4*x^3-2*x^2+4*x-1) ). - R. J. Mathar, Dec 20 2011
MATHEMATICA
f[n_] := FullSimplify[ TrigToExp[(1/4)*Sum[Sin[Pi*k/8]^2(1 + 2Cos[Pi*k/8])^n, {k, 1, 7}]]]; Table[ f[n], {n, 28}] (* Robert G. Wilson v, Jun 18 2004 *)
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Herbert Kociemba, Jun 02 2004
EXTENSIONS
a(0)=1 prepended by Alois P. Heinz, Nov 24 2023
STATUS
approved