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A124696
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Number of base-3 circular n-digit numbers with adjacent digits differing by 1 or less.
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34
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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
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OFFSET
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0,2
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COMMENTS
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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
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LINKS
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FORMULA
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[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
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)
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MAPLE
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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
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PROG
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(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))
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CROSSREFS
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KEYWORD
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nonn,base,easy
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AUTHOR
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STATUS
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approved
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