A124699 Number of base 6 circular n-digit numbers with adjacent digits differing by 1 or less.

1, 6, 16, 36, 92, 236, 622, 1658, 4468, 12132, 33146, 90998, 250802, 693426, 1922118, 5339006, 14854932, 41387764, 115438672, 322267784, 900314242, 2516648618, 7038066876, 19690060024, 55102545322, 154241612986

Offset: 0

Author

R. H. Hardin, Dec 28 2006

Keywords

• nonn
• base

Comments

[Empirical] a(base,n)=a(base-1,n)+A002426(n+1) for base>=1.int(n/2)+1

a(n) = T(n, 6) where T(n, k) = Sum_{j=1..k} (1+2*cos(j*Pi/(k+1)))^n. These are the number of smooth cyclic words of length n over the alphabet {1,2,..,6}. See theorem 3.3 in Knopfmacher and others, reference in A124696. - Peter Luschny, Aug 13 2012

Links

• OEIS Wiki, Number of base k circular n-digit numbers with adjacent digits differing by d or less

Formula

Conjectures from Colin Barker, Jun 04 2017: (Start)

G.f.: (1 - 10*x^2 + 27*x^4 - 8*x^5 - 5*x^6) / ((1 - 2*x - x^2 + x^3)*(1 - 4*x + 3*x^2 + x^3)).

a(n) = 6*a(n-1) - 10*a(n-2) + 9*a(n-4) - 2*a(n-5) - a(n-6) for n>6.

(End)