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.
LINKS
Robert Israel, Table of n, a(n) for n = 0..2174
Jim Bumgardner, Variations of the Componium, 2013.
FORMULA
From Colin Barker, Nov 26 2012: (Start)
Conjecture: a(n) = 5*a(n-1) - 6*a(n-2) - a(n-3) + 2*a(n-4) for n > 4.
G.f.: -(4*x^4 + x^3 - 12*x^2 + 3*x + 1)/((2*x - 1)*(x^3 - 3*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 8 X 8 matrix with 1 on the main diagonal and first super- and subdiagonals, 0 elsewhere, and e the column vector (1,1,1,1,1,1,1,1). Barker's conjecture follows from the fact that (A^4 - 5*A^3 + 6*A^2 + A - 2*I)*e = 0. (End)
MAPLE
f:= gfun:-rectoproc({a(n)=5*a(n-1)-6*a(n-2)-a(n-3)+2*a(n-4), a(0)=1, a(1)=8, a(2)=22, a(3)=62, a(4)=176}, a(n), remember):
map(f, [$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