A126474 - OEIS

%I #15 Jan 23 2018 21:27:58

%S 1,6,30,154,788,4034,20650,105708,541122,2770018,14179796,72586754,

%T 371573530,1902094812,9736874082,49843318162,255149275268,

%U 1306115946338,6686042370634,34226029248972,175203956722818

%N Number of arrays in [1..6]^n with adjacent elements differing by three or less.

%C [Empirical] a(base,n)=a(base-1,n)+7^(n-1) for base>=3n-2; a(base,n)=a(base-1,n)+7^(n-1)-2 when base=3n-3

%C Original name: Number of base 6 n-digit numbers with adjacent digits differing by three or less.

%H Robert Israel, <a href="/A126474/b126474.txt">Table of n, a(n) for n = 0..1408</a>

%F Conjectures from _Colin Barker_, Jan 20 2017: (Start)

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

%F G.f.: (1 + x - x^2) / (1 - 5*x - x^2 + 2*x^3).

%F (End)

%F From _Robert Israel_, Jan 23 2018: (Start)

%F a(n) = e^T M^(n-1) e where e = [1,1,1,1,1,1]^T and M is the 6 X 6 matrix with entries M(i,j) = 1 if |i-j|<=3, 0 otherwise.

%F The fact that (M^3-5*M^2-M+2I) e = 0 implies Colin Barker's recursion, and the G.f. follows. (End)

%e For n=2 the a(2)=30 solutions are [1, 1], [1, 2], [1, 3], [1, 4], [2, 1], [2, 2], [2, 3], [2, 4], [2, 5], [3, 1], [3, 2], [3, 3], [3, 4], [3, 5], [3, 6], [4, 1], [4, 2], [4, 3], [4, 4], [4, 5], [4, 6], [5, 2], [5, 3], [5, 4], [5, 5], [5, 6], [6, 3], [6, 4], [6, 5], [6, 6]. - _Robert Israel_, Jan 23 2018

%p f:= gfun:-rectoproc({a(n) = 5*a(n-1) + a(n-2) - 2*a(n-3),a(0)=1,a(1)=6,a(2)=30},a(n),remember):

%p map(f, [$0..30]); # _Robert Israel_, Jan 23 2018

%o (S/R) stvar $[N]:(0..M-1) init $[]:=0 asgn $[]->{*} kill +[i in 0..N-2](($[i]`-$[i+1]`>3)+($[i+1]`-$[i]`>3))

%Y Cf. Base 6 differing by two or less A126393, one or less A126360.

%K nonn,base

%O 0,2

%A _R. H. Hardin_, Dec 27 2006