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A059412
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Number of distinct minimal unary DFA's with exactly n states.
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7
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2, 4, 12, 30, 78, 180, 432, 978, 2220, 4926, 10908, 23790, 51750, 111564, 239604, 511758, 1089306, 2309118, 4880946, 10285146, 21619770, 45334776, 94865904, 198116082, 413013684, 859573638, 1786263822, 3706729488, 7681910826
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OFFSET
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1,1
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COMMENTS
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Also, number of binary strings w of length 2n such that the longest suffix of w appearing at least twice in w is of length n. - Jeffrey Shallit, Mar 20 2017
Also, number of ultimately periodic binary sequences uvvvv... with |u|+|v| = n, formula uses psi(|v|) and 2^(|u|-1), plus psi(n) for u empty. - Michael Vielhaber, Mar 19 2022
Also, number of minimal length-n binary patterns, each corresponding to a minimal n-state deterministic Mealy automaton outputting some binary string. Used in the definition of the Deterministic Complexity (DC) of a string w, i.e., DC(w) = n. - Lucas B. Vieira, Mar 02 2024
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REFERENCES
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M. Domaratzki, D. Kisman, and J. Shallit, On the number of distinct languages accepted by finite automata with n states, J. Autom. Lang. Combinat. 7 (2002) 4-18, Section 6, f_1(n).
Jeffrey Shallit, Notes on Enumeration of Finite Automata, manuscript, Jan 30, 2001.
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LINKS
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FORMULA
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a(n) = psi(n) + Sum_{j=1..n-1} psi(n-j)*2^(j-1), where psi(n) is the number of primitive words of length n over a 2-letter alphabet (expressible in terms of the Moebius function).
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EXAMPLE
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a(1) = 2 because there are exactly two minimal unary automata with 1 state.
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MAPLE
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end proc:
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MATHEMATICA
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A027375[n_] := DivisorSum[n, MoebiusMu[n/#]*2^# &];
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PROG
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(PARI) mypsi(n) = sumdiv(n, d, moebius(n/d)*2^d) /* from A027375 */
a(n) = mypsi(n) + sum(j=1, n-1, mypsi(n-j)*2^(j-1)) \\ Michel Marcus, May 25 2013
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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