A124698 Number of base 5 circular n-digit numbers with adjacent digits differing by 1 or less.

1, 5, 13, 29, 73, 185, 481, 1265, 3361, 8993, 24193, 65345, 177025, 480641, 1307137, 3559169, 9699841, 26452481, 72173569, 196989953, 537802753, 1468536833, 4010582017, 10954043393, 29920862209, 81733033985, 223274237953

Offset

0,2

Comments

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

a(n) = T(n, 5) 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,..,5}. See theorem 3.3 in Knopfmacher and others, reference in A124696. - Peter Luschny, Aug 13 2012

Links

Table of n, a(n) for n=0..26.

Formula

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

G.f.: (1 - 6*x^2 - 4*x^3 + 12*x^4) / ((1 - x)*(1 - 2*x)*(1 - 2*x - 2*x^2)).

a(n) = 5*a(n-1) - 6*a(n-2) - 2*a(n-3) + 4*a(n-4) for n>4.

(End)

Prog

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

Crossrefs

Sequence in context: A073857 A295112 A239367 * A337761 A147278 A268963

Adjacent sequences: A124695 A124696 A124697 * A124699 A124700 A124701

Keyword

nonn,base

Author

R. H. Hardin, Dec 28 2006

Status

approved