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A124696 Number of base-3 circular n-digit numbers with adjacent digits differing by 1 or less. 34
1, 3, 7, 15, 35, 83, 199, 479, 1155, 2787, 6727, 16239, 39203, 94643, 228487, 551615, 1331715, 3215043, 7761799, 18738639, 45239075, 109216787, 263672647, 636562079, 1536796803, 3710155683, 8957108167, 21624372015, 52205852195 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

These are the number of smooth cyclic words of length n over the alphabet {1,2,3}. See theorem 3.3 in Knopfmacher and others. - Peter Luschny, Aug 13 2012

This is the main entry for 234 similar sequences. Cf. the link to the OEIS Wiki for a list, the programs and a derivation of the linear recurrences. - Georg Fischer, Apr 09 2021

LINKS

R. H. Hardin, Table of n, a(n) for n = 0..210 [uploaded by Georg Fischer, Apr 05 2021]

Arnold Knopfmacher, Toufik Mansour, Augustine Munagi, and Helmut Prodinger, Smooth words and Chebyshev polynomials, arXiv:0809.0551v1 [math.CO], 2008.

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

Index entries for linear recurrences with constant coefficients, signature (3,-1,-1).

FORMULA

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

a(n) = T(n,3) for n > 0, where T(n,k) = Sum_{j=1..k} (1 + 2*cos(j*Pi/(k + 1)))^n. - Peter Luschny, Aug 13 2012

From Colin Barker, Nov 26 2012: (Start)

a(n) = 1 + (1 - sqrt(2))^n + (1 + sqrt(2))^n for n > 0.

a(n) = 3*a(n-1) - a(n-2) - a(n-3) for n > 3.

G.f.: -(2*x^3 + x^2 - 1)/((x - 1)*(x^2 + 2*x - 1)). (End)

MAPLE

T := (n, k) -> `if`(n=0, 1, add((1 + 2*cos(j*Pi/(k + 1)))^n, j=1..k)):

a := n -> simplify(T(n, 3)): seq(a(n), n=0..28); # Peter Luschny, Mar 28 2021

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

Cf. A002426, Row 3 of A276562.

Sequence in context: A338852 A174284 A182892 * A081669 A086821 A007576

Adjacent sequences:  A124693 A124694 A124695 * A124697 A124698 A124699

KEYWORD

nonn,base,easy

AUTHOR

R. H. Hardin, Dec 28 2006

STATUS

approved

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Last modified August 7 23:50 EDT 2022. Contains 355995 sequences. (Running on oeis4.)