A206451 Number of 0..4 arrays of length n avoiding the consecutive pattern 0..4

5, 25, 125, 625, 3124, 15615, 78050, 390125, 1950000, 9746876, 48718765, 243515775, 1217188750, 6083993750, 30410221874, 152002390605, 759768437250, 3797624997500, 18982040993750, 94879794746876, 474246971343775

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

Sequence in context: A014946 A188580 A132839 * A291164 A216126 A335506

Adjacent sequences: A206448 A206449 A206450 * A206452 A206453 A206454

Keyword

nonn

Author

R. H. Hardin, Feb 07 2012

Status

approved