cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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

Original entry on oeis.org

5, 25, 125, 625, 3124, 15615, 78050, 390125, 1950000, 9746876, 48718765, 243515775, 1217188750, 6083993750, 30410221874, 152002390605, 759768437250, 3797624997500, 18982040993750, 94879794746876, 474246971343775
Offset: 1

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Author

R. H. Hardin, Feb 07 2012

Keywords

Comments

Column 4 of A206455

Programs

  • 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

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)