OFFSET
0,2
COMMENTS
Empirical: a(base,n) = a(base-1,n) + 3^(n-1) for base >= n; a(base,n) = a(base-1,n) + 3^(n-1) - 2 when base = n-1.
Leading 0's are allowed. - Robert Israel, Aug 12 2019
LINKS
Robert Israel, Table of n, a(n) for n = 0..2231
Arnold Knopfmacher, Toufik Mansour, Augustine Munagi, and Helmut Prodinger, Smooth words and Chebyshev polynomials, arXiv:0809.0551v1 [math.CO], 2008.
FORMULA
From Colin Barker, Nov 26 2012: (Start)
Conjecture: a(n) = 4*a(n-1) - 3*a(n-2) - a(n-3) for n > 3.
G.f.: -(x^3 + 5*x^2 - 2*x - 1)/(x^3 + 3*x^2 - 4*x + 1). (End)
From Robert Israel, Aug 12 2019: (Start)
a(n) = e^T A^(n-1) e for n >= 1, where A is the 6 X 6 matrix with 1 on the main diagonal and first super- and subdiagonals, 0 elsewhere, and e the column vector (1,1,1,1,1,1).
Barker's conjecture follows from the fact that (A^3 - 4*A^2 + 3*A + 1) e = 0. (End)
MAPLE
A:=LinearAlgebra:-ToeplitzMatrix([1, 1, 0, 0, 0, 0], symmetric):
e:= Vector(6, 1):
1, seq(e^%T . A^n . e, n=0..30); # Robert Israel, Aug 12 2019
PROG
(S/R) stvar $[N]:(0..M-1) init $[]:=0 asgn $[]->{*} kill +[i in 0..N-2](($[i]`-$[i+1]`>1)+($[i+1]`-$[i]`>1))
CROSSREFS
KEYWORD
nonn,base
AUTHOR
R. H. Hardin, Dec 26 2006
STATUS
approved