A183356 - OEIS

%I #15 Apr 22 2018 04:53:10

%S 36,576,1296,3600,9216,24336,63504,166464,435600,1140624,2985984,

%T 7817616,20466576,53582400,140280336,367258896,961496064,2517229584,

%U 6590192400,17253347904,45169851024,118256205456,309598765056,810540090000

%N One quarter the number of n X 4 1..4 arrays with no two neighbors of any element equal to each other.

%C Column 4 of A183362.

%H R. H. Hardin, <a href="/A183356/b183356.txt">Table of n, a(n) for n = 1..200</a>

%F Empirical: a(n) = 2*a(n-1) + 2*a(n-2) - a(n-3) for n>4.

%F Conjectures from _Colin Barker_, Mar 28 2018: (Start)

%F G.f.: 36*x*(1 + 14*x + 2*x^2 - 3*x^3) / ((1 + x)*(1 - 3*x + x^2)).

%F a(n) = (9/5)*2^(3-n)*((-1)^n*2^(2+n) + (3-sqrt(5))^(1+n) + (3+sqrt(5))^(1+n)) for n>1.

%F (End)

%F Assuming Colin Barker's conjectures, a(n) = (12*Fibonacci(n+1))^2, n>1. - _Ehren Metcalfe_, Apr 21 2018

%e Some solutions for 5 X 4 with a(1,1)=1:

%e   1 4 3 3    1 1 4 4    1 2 4 1    1 4 2 2    1 1 3 4

%e   2 4 1 1    4 2 3 1    3 2 4 1    1 4 3 3    2 2 3 1

%e   3 3 2 2    4 2 3 1    4 1 3 3    3 2 1 1    3 4 4 2

%e   1 1 4 4    3 1 4 4    4 1 2 2    3 2 4 4    3 1 1 3

%e   4 2 3 1    3 1 2 2    2 3 4 1    1 1 3 2    4 2 2 4

%Y Cf. A183362.

%K nonn

%O 1,1

%A _R. H. Hardin_, Jan 04 2011