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A349002
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The number of Lyndon words of size n from an alphabet of 4 letters and 1st, 2nd and 3rd letter of the alphabet with equal frequency in the words.
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1
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1, 1, 0, 2, 6, 12, 34, 120, 354, 1082, 3636, 12270, 40708, 139062, 484866, 1692268, 5944470, 21134808, 75625330, 271720146, 982116648, 3569558058, 13025614962, 47714385708, 175470892468, 647508620070, 2396613522804, 8896422981608, 33114570409896, 123566641829256
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OFFSET
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0,4
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COMMENTS
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Counts a subset of the Lyndon words in A027377. Here there is no requirement of how often the 4th letter of the alphabet occurs in the admitted word, only on the frequency of the 1st to 3rd letter of the alphabet.
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LINKS
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FORMULA
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EXAMPLE
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Examples for the alphabet {0,1,2,3}:
a(0)=1 counts (), the empty word.
a(3)=2 counts (021) (012).
a(4)=6 counts (0321) (0231) (0312) (0132) (0213) (0123).
a(5)=12 counts (03321) (03231) (02331) (03312) (03132) (01332) (03213) (02313) (03123) (01323) (02133) (01233).
a(6)=34 counts (020211) (002211) (012021) (002121) (010221) (001221) (033321) (033231) (032331) (023331) (012102) (011202) (002112) (010212) (001212) (033312) (011022) (010122) (001122) (033132) (031332) (013332) (033213) (032313) (023313) (033123) (031323) (013323) (032133) (023133) (031233) (013233) (021333) (012333).
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PROG
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(PARI) a(n) = if(n>0, sumdiv(n, d, moebius(n/d)*sum(k=0, d\3, d!/(k!^3*(d-3*k)!)))/n, n==0) \\ Andrew Howroyd, Jan 14 2023
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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