cp's OEIS Frontend

A126362 Number of base 8 n-digit numbers with adjacent digits differing by one or less.

%I A126362 #19 Jan 13 2022 05:07:58
%S A126362 1,8,22,62,176,502,1436,4116,11814,33942,97582,280676,807574,2324116,
%T A126362 6689624,19257202,55439298,159611886,459545688,1323132230,3809653732,
%U A126362 10969153364,31583803574,90940708414,261850874726,753964626300
%N A126362 Number of base 8 n-digit numbers with adjacent digits differing by one or less.
%C A126362 [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.
%H A126362 Robert Israel, <a href="/A126362/b126362.txt">Table of n, a(n) for n = 0..2174</a>
%H A126362 Jim Bumgardner, <a href="http://jbum.com/papers/componium_variations.pdf">Variations of the Componium</a>, 2013.
%F A126362 From _Colin Barker_, Nov 26 2012: (Start)
%F A126362 Conjecture: a(n) = 5*a(n-1) - 6*a(n-2) - a(n-3) + 2*a(n-4) for n > 4.
%F A126362 G.f.: -(4*x^4 + x^3 - 12*x^2 + 3*x + 1)/((2*x - 1)*(x^3 - 3*x + 1)). (End)
%F A126362 From _Robert Israel_, Aug 12 2019: (Start)
%F A126362 a(n) = e^T A^(n-1) e for n>=1, where A is the 8 X 8 matrix with 1 on the main diagonal and first super- and subdiagonals, 0 elsewhere, and e the column vector (1,1,1,1,1,1,1,1). Barker's conjecture follows from the fact that (A^4 - 5*A^3 + 6*A^2 + A - 2*I)*e = 0. (End)
%p A126362 f:= gfun:-rectoproc({a(n)=5*a(n-1)-6*a(n-2)-a(n-3)+2*a(n-4),a(0)=1,a(1)=8,a(2)=22,a(3)=62,a(4)=176},a(n),remember):
%p A126362 map(f, [$0..30]); # _Robert Israel_, Aug 12 2019
%o A126362 (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 A126362 nonn,base
%O A126362 0,2
%A A126362 _R. H. Hardin_, Dec 26 2006