A006690 -id:A006690 - cp's OEIS frontend

A006689 Number of deterministic, completely-defined, initially-connected finite automata with 2 inputs and n unlabeled states.

1, 12, 216, 5248, 160675, 5931540, 256182290, 12665445248, 705068085303, 43631250229700, 2970581345516818, 220642839342906336, 17753181687544516980, 1538156947936524172656, 142767837727544113783650, 14132742269814671677890048, 1486239549269418316509918111, 165468157149718534883789004804

Offset: 1

N. J. A. Sloane

a(n) is divisible by n^2, see A082166. These automata have no nontrivial automorphisms (by states).
Equals the first column of triangle A107670, which is the matrix square of triangle A107667. As a lower triangular matrix T, A107667 satisfies: T = D + SHIFT_LEFT(T^2) where SHIFT_LEFT shifts each row 1 place left and D is the diagonal matrix [1,2,3,...]. - Paul D. Hanna, May 19 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

			a(2) = 12 since the following transition diagrams represent all twelve initially connected automata with two input letters x and y and two states 1 (initial) and 2: 1==x,y==>2==>, 1--x-->2==>, 1--y-->2==>, 1--y-->1 1--x-->1 where the transitions from state 2 are specified arbitrary (4 different possibilities in every case).

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
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).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

M. Almeida, N. Moreira and R. Reis. On the Representation of Finite Automata, Technical Report DCC-2005-04, DCC - FC & LIACC, Universidade do Porto, April, 2005.
M. Almeida, N. Moreira, and R. Reis, Enumeration and generation with a string automata representation, Theor. Comp. Sci. 387 (2007) 93-102, B(k=2,n)
E. Lebensztayn, A large deviations principle for the Maki-Thompson rumour model, arXiv preprint arXiv:1411.5614 [math.PR], 2014-2015.
V. A. Liskovets, The number of initially connected automata, Kibernetika, (Kiev), No3 (1969), 16-19; Engl. transl.: Cybernetics, v.4 (1969), 259-262.
V. A. Liskovets, Exact enumeration of acyclic automata, Proc. 15th Conf. "Formal Power Series and Algebr. Combin. (FPSAC'03)", 2003.
V. A. Liskovets, Exact enumeration of acyclic deterministic automata, Discrete Appl. Math., 154, No.3 (2006), 537-551.
R. 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.]
G. Sedlitz, Abzahlung von Automaten, formalen Sprachen und verwandten Strukturen, Master's Thesis, Vienna (2017), Theorem 6.1

Cf. A006690, A107670, A107667.

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:
A006689 := proc(n)
    B(2,n) ;
end proc:
seq(A006689(n),n=1..10) ; # R. J. Mathar, May 21 2018

a[1] = 1; a[n_] := a[n] = n^(2*n)/(n-1)! - Sum[n^(2*(n-i))*a[i]/(n-i)!, {i, 1, n-1}]; Table[ a[n], {n, 1, 15}] (* Jean-François Alcover, Dec 15 2014 *)

a(n)=if(n<1,0,n^(2*n)/(n-1)!-sum(i=1,n-1,n^(2*(n-i))/(n-i)!*a(i)))

a(n)=local(A);if(n<1,0,A=n*x+x*O(x^n); for(k=0,n,A+=polcoeff(A,k)*x^k*(n-prod(i=0,k,1-(n-i)*x)));polcoeff(A,n))

a(n) = h_2(n)/(n-1)!, where h_2(1) := 1, h_2(n) := n^(2*n) - Sum_{i=1..n-1} binomial(n-1, i-1)*n^(2*n-2*i)*h_2(i) for n > 1.

For k = 2, 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, Feb 26 2021. See Theorem 8 in Almeida, Moreira, and Reis (2007). The value of f_0 is not relevant.]

More terms and more detailed definition from Valery A. Liskovets, Apr 09 2003
Further terms from Paul D. Hanna, May 19 2005
Edited by N. J. A. Sloane, Dec 06 2008 at the suggestion of R. J. Mathar

A107671 Triangular matrix T, read by rows, that satisfies: T = D + SHIFT_LEFT(T^3), where SHIFT_LEFT shifts each row 1 place to the left and D is the diagonal matrix {1, 2, 3, ...}.

Original entry on oeis.org

1, 8, 2, 513, 27, 3, 81856, 2368, 64, 4, 23846125, 469625, 7625, 125, 5, 10943504136, 160767720, 1898856, 19656, 216, 6, 7250862593527, 83548607478, 776598305, 6081733, 43561, 343, 7, 6545029128786432, 61068815111168, 465690017280, 2966844928, 16494080, 86528, 512, 8

Offset: 0

Paul D. Hanna, Jun 07 2005

			Triangle T begins:
              1;
              8,           2;
            513,          27,         3;
          81856,        2368,        64,       4;
       23846125,      469625,      7625,     125,     5;
    10943504136,   160767720,   1898856,   19656,   216,   6;
  7250862593527, 83548607478, 776598305, 6081733, 43561, 343, 7;
  ...
The matrix cube T^3 shifts each row to the right 1 place, dropping the diagonal D and putting A006690 in column 0:
             1;
            56,           8;
          7965,         513,        27;
       2128064,       81856,      2368,      64;
     914929500,    23846125,    469625,    7625,  125;
  576689214816, 10943504136, 160767720, 1898856, 19656, 216;
  ...
From _Petros Hadjicostas_, Mar 11 2021: (Start)
We illustrate the above formula for T(n,k=0) with the compositions of n + 1 for n = 2. The compositions of n + 1 = 3 are 3, 1 + 2, 2 + 1, and 1 + 1 + 1.  Thus the above sum has four terms with (r = 1, s_1 = 3), (r = 2, s_1 = 1, s_2 = 2), (r = 2, s_1 = 2, s_2 = 1), and (r = 3, s_1 = s_2 = s_3 = 1).
The value of the denominator Product_{j=1..r} s_j! for these four terms is 6, 2, 2, and 1, respectively.
The value of the numerator s_1^(-1)*Product_{j=1..r} (Sum_{i=1..j} s_i)^(3*s_j) for these four terms is 19683/3, 729/1, 1728/2, and 216/1.
Thus T(2,0) = (19683/3)/6 - (729/1)/2 - (1728/2)/2 + (216/1)/1 = 513. (End)

Cf. A107667, A107672 (column 0), A107673, A107674 (matrix square), A107676 (matrix cube), A006690.

{T(n,k)=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)));if(n>=k,(P^-1*D*P)[n+1,k+1])}

Matrix diagonalization method: define the triangular matrix P by P(n, k) = ((n+1)^3)^(n-k)/(n-k)! for n >= k >= 0 and the diagonal matrix D(n, n) = n+1 for n >= 0; then T is given by T = P^-1*D*P.

T(n,k=0) = Sum_{r=1..(n+1)} (-1)^(r-1) * Sum_{s_1, ..., s_r} (s_1^(-1)/(Product_{j=1..r} s_j!)) * Product_{j=1..r} (Sum_{i=1..j} s_i)^(3*s_j)), where the second sum is over lists (s_1, ..., s_r) of positive integers s_i such that Sum_{i=1..r} s_i = n + 1. (Thus, the second sum is over all compositions of n + 1.) - Petros Hadjicostas, Mar 11 2021

A107676 Matrix cube of triangle A107671.

Original entry on oeis.org

1, 56, 8, 7965, 513, 27, 2128064, 81856, 2368, 64, 914929500, 23846125, 469625, 7625, 125, 576689214816, 10943504136, 160767720, 1898856, 19656, 216, 500750172337212, 7250862593527, 83548607478, 776598305, 6081733, 43561, 343

Offset: 0

Paul D. Hanna, Jun 07 2005

Column 0 is A006690.

			Triangle begins:
1;
56,8;
7965,513,27;
2128064,81856,2368,64;
914929500,23846125,469625,7625,125;
576689214816,10943504136,160767720,1898856,19656,216; ...
which equals the matrix cube of triangle A107671:
1;
8,2;
513,27,3;
81856,2368,64,4;
23846125,469625,7625,125,5;
10943504136,160767720,1898856,19656,216,6; ...

Cf. A107667, A107671, A107674, A006690.

{T(n,k)=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)));if(n>=k,(P^-1*D^3*P)[n+1,k+1])}

Matrix diagonalization method: define triangular matrix P by P(n, k) = ((n+1)^3)^(n-k)/(n-k)!, n>=k>=0 and diagonal matrix D(n, n) = n+1, then T is given by T = P^-1*D^3*P.

A107674 Matrix square of triangle A107671.

Original entry on oeis.org

1, 24, 4, 2268, 135, 9, 461056, 15936, 448, 16, 160977375, 3789250, 69000, 1125, 25, 85624508376, 1485395280, 19994688, 223560, 2376, 36, 64363893844726, 862907827866, 9138674195, 79086196, 596820, 4459, 49, 64928246784463872

Offset: 0

Paul D. Hanna, Jun 07 2005

Column 0 is A107675.

			Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:
            1;
           24,          4;
         2268,        135,        9;
       461056,      15936,      448,     16;
    160977375,    3789250,    69000,   1125,   25;
  85624508376, 1485395280, 19994688, 223560, 2376, 36;
  ...

Cf. A006690, A107667, A107671, A107675, A107676.

{T(n,k)=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)));if(n>=k,(P^-1*D^2*P)[n+1,k+1])}

Matrix diagonalization method: define the triangular matrix P by P(n, k) = ((n+1)^3)^(n-k)/(n-k)! for n >= k >= 0 and the diagonal matrix D by D(n, n) = n+1 for n >= 0; then T is given by T = P^-1*D^2*P.

A342234 a(n) = (27^n - 9^n)/2 - 12^n + 6^n.

Original entry on oeis.org

0, 3, 216, 7965, 243000, 6903873, 190505196, 5192233245, 140764942800, 3807455329593, 102881965757076, 2778771947174325, 75038262510065400, 2026169325431888913, 54708199287259567356, 1477140843778461200205, 39883035730488375376800, 1076844754605007952329833

Offset: 0

Petros Hadjicostas, Mar 06 2021

For n >= 1, a(n) is the number of deterministic, completely-defined, initially-connected finite automata with n inputs and 3 unlabeled states. A020522 counts similar automata with n inputs and 2 unlabeled states.
According to a comment by Nelma Moreira in A006689 and A006690, the number of such automata with N inputs and M unlabeled states is Sum (Product_{i=1..M-1} i^(f_i - f_{i-1} - 1)) * M^(M*N - f_{M-1} - 1), where the sum is taken over integers f_1, ..., f_{M-1} satisfying 0 <= f_1 < N and f_{i-1} < f_{i} < i*N for i = 2..M-1. (See Theorem 8 in Almeida, Moreira, and Reis (2007). The value of f_0 is not relevant.) For this sequence we have M = 3 unlabeled states, for A020522 we have M = 2 unlabeled states, for A006689 we have N = 2 inputs, and for A006690 we have N = 3 inputs.
A similar formula for the number of such automata with N inputs and M unlabeled states was given by Robinson (1985, Eq. (2.3) upon division by (p-1)!). It is Sum_{r=1..M} (-1)^(r-1) * Sum_{k_1,...,k_r} (k_1/(Product_{i=1..r} k_i!)) * Product_{j=1..r} (Sum_{i=1..j} k_i)^(N*k_j), where the second sum is over all compositions (k_1,...,k_r) of length r of M. (A composition of length r of M is an ordered partition (k_1,...,k_r) with k_i >= 1 for i = 1..r and Sum_{i=1..r} k_i = M.)
Both formulas with N = n and M = 3 give a(n) = (27^n - 9^n)/2 - 12^n + 6^n.
In Liskovets (2006, Eq. (11), p. 546), a(n) equals H_N(M) = h_N(M)/(M-1)! with N = n and M = 3. The sequence h_N(M) satisfies the recurrence h_N(M) = M^(N*M) - Sum_{t=1..M-1} binomial(M-1, t-1) * M^(N*(M-t)) * h_N(t) with h_N(1) = 1. A recurrence for  H_N(M) = h_N(M)/(M-1)! originally appeared in Liskovets (1969, Eq. (3), p. 17, denoted by h_{n,r}).

M. Almeida, N. Moreira, and R. Reis, Enumeration and generation with a string automata representation, Theor. Comp. Sci. 387 (2007), 93-102; see B(k=3,n).
Valery A. Liskovets, The number of connected initial automata, Kibernetika (Kiev), 3 (1969), 16-19 (in Russian; English translation: Cybernetics, 4 (1969), 259-262).
Valery A. Liskovets, Exact enumeration of acyclic automata, Proc. 15th Conf. "Formal Power Series and Algebr. Combin. (FPSAC'03)", 2003.
Valery A. Liskovets, Exact enumeration of acyclic deterministic automata, Discrete Appl. Math., 154, No. 3 (2006), 537-551.
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. [Annotated scanned copy, with permission of the author.]
Index entries for linear recurrences with constant coefficients, signature (54,-963,6966,-17496).

Cf. A006689, A006690, A020522.

lista(nn) = {my(h=matrix(nn+3,nn+3)); my(H=vector(nn+1)); for(N=1, nn, for(M=1, nn, h[N,M] = if(M==1, 1,  M^(N*M)-sum(t=1,M-1, binomial(M-1, t-1)*M^(N*(M-t))*h[N,t]))));
for(N=1, nn+1, H[N] = if(N==1, 0, h[N-1,3]/2)); H;}

G.f.: 3*x*(1 + 18*x - 270*x^2)/(1 - 54*x + 963*x^2 - 6966*x^3 + 17496*x^4). - Stefano Spezia, Mar 08 2021

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A082168 a(n) = A006690(n)/n^3.

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A082167 Duplicate of A006690.

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A006689 Number of deterministic, completely-defined, initially-connected finite automata with 2 inputs and n unlabeled states.

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A107671 Triangular matrix T, read by rows, that satisfies: T = D + SHIFT_LEFT(T^3), where SHIFT_LEFT shifts each row 1 place to the left and D is the diagonal matrix {1, 2, 3, ...}.

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A107676 Matrix cube of triangle A107671.

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A107674 Matrix square of triangle A107671.

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A342234 a(n) = (27^n - 9^n)/2 - 12^n + 6^n.

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