login
Number of length-n 0..2 arrays with no repeated value equal to the previous repeated value.
2

%I #7 Mar 21 2018 17:20:03

%S 3,9,24,66,174,462,1206,3150,8166,21150,54582,140718,362118,931134,

%T 2391894,6141006,15757734,40420062,103647606,265721070,681097926,

%U 1745555070,4473092502,11461604238,29366557158,75238139934,192754700214

%N Number of length-n 0..2 arrays with no repeated value equal to the previous repeated value.

%C Column 2 of A269467.

%H R. H. Hardin, <a href="/A269461/b269461.txt">Table of n, a(n) for n = 1..210</a>

%F Empirical: a(n) = 3*a(n-1) + 2*a(n-2) - 8*a(n-3).

%F Conjectures from _Colin Barker_, Mar 21 2018: (Start)

%F G.f.: 3*x*(1 - 3*x^2) / ((1 - 2*x)*(1 - x - 4*x^2)).

%F a(n) = 2^(-4-n)*(-51*4^(1+n) + (255-57*sqrt(17))*(1-sqrt(17))^n + 3*(1+sqrt(17))^n*(85+19*sqrt(17))) / 17.

%F (End)

%e Some solutions for n=9:

%e ..1. .1. .0. .2. .1. .0. .1. .2. .2. .1. .1. .0. .2. .2. .0. .2

%e ..1. .2. .1. .2. .2. .1. .0. .1. .1. .0. .1. .2. .0. .1. .1. .1

%e ..0. .2. .2. .1. .1. .2. .1. .1. .0. .0. .0. .1. .1. .1. .0. .1

%e ..2. .0. .2. .2. .2. .2. .2. .0. .0. .1. .2. .2. .1. .0. .1. .2

%e ..2. .1. .0. .0. .0. .1. .2. .2. .2. .2. .0. .0. .2. .0. .0. .0

%e ..1. .0. .2. .1. .1. .0. .0. .2. .1. .2. .0. .1. .1. .2. .2. .0

%e ..2. .1. .1. .1. .0. .2. .2. .1. .1. .0. .1. .0. .0. .2. .1. .2

%e ..0. .1. .2. .0. .0. .1. .1. .0. .0. .2. .2. .1. .2. .1. .1. .2

%e ..1. .0. .1. .0. .1. .0. .0. .1. .1. .1. .2. .0. .0. .1. .2. .0

%Y Cf. A269467.

%K nonn

%O 1,1

%A _R. H. Hardin_, Feb 27 2016