OFFSET
1,1
COMMENTS
Column 4 of A206455
LINKS
R. H. Hardin, Table of n, a(n) for n = 1..210
Index entries for linear recurrences with constant coefficients, signature (5,0,0,0,-1).
FORMULA
a(n) = 5*a(n-1) -a(n-5)
Empirical: a(n) = sum{i in 0..floor(n/5)} ((-1)^i*5^(n-5*i)*binomial(n-4*i,i))
From Robert Israel, Jan 08 2016: (Start) The recursion can be proved using the matrix representation
a(n) = [ 1 1 1 1 1] M^n [ 1 0 0 0 0 ]^T, where
M = [ 4 3 3 3 3 ]
[ 1 1 1 1 1 ]
[ 0 1 0 0 0 ]
[ 0 0 1 0 0 ]
[ 0 0 0 1 0 ]
which satisfies M^5 = 5 M^4 - I.
G.f.: -x*(-5+x^4) / ( 1-5*x+x^5 ).. (End)
MAPLE
M:= <<4|3|3|3|3>, <1|1|1|1|1>, <0|1|0|0|0>, <0|0|1|0|0>, <0|0|0|1|0>>:
seq(<1|1|1|1|1> . M^n . <1, 0, 0, 0, 0>, n=1..30); # Robert Israel, Jan 08 2016
CROSSREFS
KEYWORD
nonn
AUTHOR
R. H. Hardin, Feb 07 2012
STATUS
approved