A107676 -id:A107676 - cp's OEIS frontend

A006690 Number of deterministic, completely-defined, initially-connected finite automata with 3 inputs and n unlabeled states.

1, 56, 7965, 2128064, 914929500, 576689214816, 500750172337212, 572879126392178688, 835007874759393878655, 1510492370204314777345000, 3320470273536658970739763334, 8718034433102107344888781813632, 26945647825926481227016730431025962, 96843697086370972449408988324175689680

Offset: 1

N. J. A. Sloane

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

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).

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, 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. xv+810 pp. ISBN: 0-471-81635-3; Math Review MR0812651. (86g:05026). [Annotated scanned copy, with permission of the author.]

Cf. A006689, A006689, A107671, A107676.

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:
A006690 := proc(n)
    B(3,n) ;
end proc:
seq(A006690(n),n=1..10) ; # R. J. Mathar, May 21 2018

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 *)

{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

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

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.]

a(11) and more detailed definition from Valery A. Liskovets, Apr 09 2003
Edited by N. J. A. Sloane, Dec 06 2008 at the suggestion of R. J. Mathar
More terms from M. F. Hasler, May 16 2018

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

A107675 Column 0 of triangle A107674.

Original entry on oeis.org

1, 24, 2268, 461056, 160977375, 85624508376, 64363893844726, 64928246784463872, 84623205378726331245, 138408056280920732755000, 277597038523589348539241112, 670011760601512512626484887040

Offset: 0

Paul D. Hanna, Jun 07 2005

nonn

The o.g.f. A(x) = Sum_{m >= 0} a(m)*x^m is such that, for each integer n > 0, the coefficient of x^n in the expansion of exp(n^3*x)*(1 - x*A(x)) is equal to 0.

Given the o.g.f. A(x), the o.g.f. of A304323 equals 1/(1 - x*A(x)).

			O.g.f.: A(x) = 1 + 24*x + 2268*x^2 + 461056*x^3 + 160977375*x^4 + 85624508376*x^5 + 64363893844726*x^6 + 64928246784463872*x^7 + ...

Cf. A107671, A107674, A107676.

Cf. A304323, A107668, A304394, A304395, A342202.

{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^2*P)[n+1,1]}
for(n=0,20, print1(a(n),", "))

/* From formula: [x^n] exp( n^3*x ) * (1 - x*A(x)) = 0 */
{a(n) = my(A=[1]); for(i=0, n, A=concat(A, 0); m=#A; A[m] = Vec( exp(x*m^3 +x^2*O(x^m)) * (1 - x*Ser(A)) )[m+1] ); A[n+1]}
for(n=0, 25, print1( a(n), ", ")) \\ Paul D. Hanna, May 12 2018

/* From Recurrence: */
{a(n) = if(n==0,1, (n+1)^(3*n+3)/(n+1)! - sum(k=1,n, (n+1)^(3*k)/k! * a(n-k) ))}
for(n=0,25, print1( a(n),", ")) \\ Paul D. Hanna, May 12 2018

O.g.f. A(x) satisfies: [x^n] exp(n^3*x) * (1 - x*A(x)) = 0 for n > 0. - Paul D. Hanna, May 12 2018
a(n) = (n+1)^(3*n+3)/(n+1)! - Sum_{k=1..n} (n+1)^(3*k)/k! * a(n-k) for n > 0 with a(0) = 1. - Paul D. Hanna, May 12 2018
a(n) = A342202(3,n+1) = Sum_{r=1..(n+1)} (-1)^(r-1) * Sum_{s_1, ..., s_r} (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. See Michel Marcus's PARI program in A342202.) - Petros Hadjicostas, Mar 10 2021

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.

A107672 Column 0 of triangle A107671.

Original entry on oeis.org

1, 8, 513, 81856, 23846125, 10943504136, 7250862593527, 6545029128786432, 7720335872745730749, 11531675416606553251000, 21278751956820661358187902, 47547062997060115956475702656, 126548714317113405123981003974183

Offset: 0

Paul D. Hanna, Jun 07 2005

nonn

Cf. A107671, A107673, A107676.

{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*P)[n+1,1]}

a(n) = (n+1)^3*A107673(n). [Corrected by Petros Hadjicostas, Mar 11 2021]

a(n) = 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

cp's OEIS Frontend

A006690 Number of deterministic, completely-defined, initially-connected finite automata with 3 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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A107675 Column 0 of triangle A107674.

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A107674 Matrix square of triangle A107671.

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A107672 Column 0 of triangle A107671.

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A107673 a(n) = A107672(n)/(n+1)^3.

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