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 A206451 Number of 0..4 arrays of length n avoiding the consecutive pattern 0..4 2
 5, 25, 125, 625, 3124, 15615, 78050, 390125, 1950000, 9746876, 48718765, 243515775, 1217188750, 6083993750, 30410221874, 152002390605, 759768437250, 3797624997500, 18982040993750, 94879794746876, 474246971343775 (list; graph; refs; listen; history; text; internal format)
 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

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Last modified September 24 03:26 EDT 2021. Contains 347623 sequences. (Running on oeis4.)