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A126360 Number of base 6 n-digit numbers with adjacent digits differing by one or less. 7

%I

%S 1,6,16,44,122,340,950,2658,7442,20844,58392,163594,458356,1284250,

%T 3598338,10082246,28249720,79153804,221783810,621424108,1741191198,

%U 4878708658,13669836930,38302030548,107319902744,300703682402

%N Number of base 6 n-digit numbers with adjacent digits differing by one or less.

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

%C Leading 0's are allowed. - _Robert Israel_, Aug 12 2019

%H Robert Israel, <a href="/A126360/b126360.txt">Table of n, a(n) for n = 0..2231</a>

%H Arnold Knopfmacher, Toufik Mansour, Augustine Munagi, Helmut Prodinger, <a href="http://arxiv.org/abs/0809.0551">Smooth words and Chebyshev polynomials</a>, arXiv:0809.0551v1 [math.CO], 2008.

%F Conjecture: a(n) = 4*a(n-1)-3*a(n-2)-a(n-3) for n>3. G.f.: -(x^3+5*x^2-2*x-1)/(x^3+3*x^2-4*x+1). [_Colin Barker_, Nov 26 2012]

%F From _Robert Israel_, Aug 12 2019: (Start)

%F a(n) = e^T A^(n-1) e for n>=1, where A is the 6 X 6 matrix with 1 on the main diagonal and first super- and sub-diagonals, 0 elsewhere, and e the column vector (1,1,1,1,1,1).

%F Barker's conjecture follows from the fact that (A^3-4*A^2+3*A+1) e = 0. (End)

%p A:=LinearAlgebra:-ToeplitzMatrix([1,1,0,0,0,0],symmetric):

%p e:= Vector(6,1):

%p 1, seq(e^%T . A^n . e, n=0..30); # _Robert Israel_, Aug 12 2019

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

%K nonn,base

%O 0,2

%A _R. H. Hardin_, Dec 26 2006

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Last modified March 30 22:55 EDT 2020. Contains 333132 sequences. (Running on oeis4.)