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A006690
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Number of deterministic, completely-defined, initially-connected finite automata with 3 inputs and n unlabeled states.
(Formerly M5316)
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8
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1, 56, 7965, 2128064, 914929500, 576689214816, 500750172337212, 572879126392178688, 835007874759393878655, 1510492370204314777345000, 3320470273536658970739763334, 8718034433102107344888781813632, 26945647825926481227016730431025962, 96843697086370972449408988324175689680
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OFFSET
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1,2
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COMMENTS
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a(n) is divisible by n^3, see A082168. These automata have no nontrivial automorphisms (by states).
Found in column 0 of triangle A107676, which is the matrix cube of triangle A107671 (see recurrence formulas). - Paul D. Hanna, Jun 07 2005
A complete initially connected deterministic finite automaton (icdfa) with n states in an alphabet of k symbols can be represented by a special string of {0,...,n-1}^* with length kn. In that string, let f_i be the index of the first occurrence of state i (used in the formula). - Nelma Moreira, Jul 31 2005
This is H_3(n) in Liskovets (DAM, Vol. 154, 2006), p. 548; the formula for H_k is given in Eq. (11), p. 546. - M. F. Hasler, May 16 2018
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REFERENCES
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R. Bacher and C. Reutenauer, The number of right ideals of given codimension over a finite field, in Noncommutative Birational Geometry, Representations and Combinatorics, edited by Arkady. Berenstein and Vladimir. Retakha, Contemporary Mathematics, Vol. 592, 2013.
V. A. Liskovets, The number of initially connected automata, Kibernetika, (Kiev), No3 (1969), 16-19; Engl. transl.: Cybernetics, v.4 (1969), 259-262.
R. Reis, N. Moreira and M. Almeida, On the Representation of Finite Automata, in Proocedings of 7th Int. Workshop on Descriptional Complexity of Formal Systems (DCFS05) Jun 30, 2005, Como, Italy, page 269-276
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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Robert W. Robinson, Counting strongly connected finite automata, pages 671-685 in "Graph theory with applications to algorithms and computer science." Proceedings of the fifth international conference held at Western Michigan University, Kalamazoo, Mich., June 4-8, 1984. Edited by Y. Alavi, G. Chartrand, L. Lesniak [L. M. Lesniak-Foster], D. R. Lick and C. E. Wall. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1985. xv+810 pp. ISBN: 0-471-81635-3; Math Review MR0812651. (86g:05026). [Annotated scanned copy, with permission of the author.]
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FORMULA
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a(n) = h_3(n)/(n-1)!, where h_3(1) := 1, h_3(n) := n^(3*n) - Sum_{i=1..n-1} binomial(n-1, i-1) * n^(3*n-3*i) * h_3(i) for n > 1.
For k = 3, a(n) = Sum (Product_{i=1..n-1} i^(f_i - f_{i-1} - 1))) * n^(n*k - f_{n-1} - 1), where the sum is taken over integers f_1, ..., f_{n-1} satisfying 0 <= f_1 < k and f_{i-1} < f_{i} < i*k for i = 2..n-1. - Nelma Moreira, Jul 31 2005 [Typo corrected by Petros Hadjicostas, Mar 06 2021. See Theorem 8 in Almeida, Moreira, and Reis (2007). The value of f_0 is not relevant.]
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MAPLE
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b := proc(k, n)
option remember;
if n = 1 then
1;
else
n^(k*n) -add(binomial(n-1, j-1)*n^(k*(n-j))*procname(k, j), j=1..n-1) ;
end if;
end proc:
B := proc(k, n)
b(k, n)/(n-1)! ;
end proc:
B(3, n) ;
end proc:
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MATHEMATICA
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a[1] = 1; a[n_] := a[n] = n^(3*n)/(n-1)! - Sum[n^(3*(n-i))*a[i]/(n-i)!, {i, 1, n-1}]; Table[a[n], {n, 1, 11}] (* Jean-François Alcover, Dec 15 2014 *)
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PROG
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(PARI) {a(n)=local(P=matrix(n+1, n+1, r, c, if(r>=c, (r^3)^(r-c)/(r-c)!)), D=matrix(n+1, n+1, r, c, if(r==c, r))); (P^-1*D^3*P)[n+1, 1]} \\ Paul D. Hanna, Jun 07 2005
(PARI) A6690=[1]; A006690(n)={for(n=#A6690+1, n, A6690=concat(A6690, n^(3*n)/(n-1)!-sum(k=1, n-1, n^(3*k)*A6690[n-k]/k!))); A6690[n]} \\ M. F. Hasler, May 16 2018
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CROSSREFS
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KEYWORD
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easy,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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