A277839 -id:A277839 - cp's OEIS frontend

A277844 Number of finite automata with n states, n input symbols and six output symbols.

1, 81, 11328242, 29902254119865429, 817701868164546859278494745163, 285033249600409431428643990739291312182972084132, 1892438067444572851650149500498661434054764424790064535313952779756847, 339676614862729029614552301296020122485910436927008569295805654935518977116532247635480871741432

Offset: 1

Marko Riedel, Nov 01 2016

nonn

Scott Burns, Marko Riedel, Markus Scheuer, Number of functions, Math StackExchange.
Scott Burns, Marko Riedel, Markus Scheuer, Enumeration of finite automata, Math StackExchange.
F. Harary and E. Palmer, Enumeration of finite automata, Information and Control, 10 (1967), 499-508.
Marko Riedel, Maple code for A277839-A277844.

Cf. A277839, A277840, A277841, A277842, A277843.

A277840 Number of finite automata with n states, n input symbols and two output symbols.

Original entry on oeis.org

1, 44, 142336, 244698934716, 347256965617453111707, 683647218221456315461840833799588, 2846969183281612697167894035560332610102537605107, 35612941825082950044316879351953518880328546726186269125209259942000, 1805164781998352977708541832242375664226097624365740577986086562708182817353463604362494621

Offset: 1

Marko Riedel, Nov 01 2016

nonn

Scott Burns, Marko Riedel, Markus Scheuer, Number of functions, Math StackExchange.
Scott Burns, Marko Riedel, Markus Scheuer, Enumeration of finite automata, Math StackExchange.
F. Harary and E. Palmer, Enumeration of finite automata, Information and Control, 10 (1967), 499-508.
Marko Riedel, Maple code for A277839-A277844.

Cf. A277839, A277841, A277842, A277843, A277844.

A277841 Number of finite automata with n states, n input symbols and three output symbols.

Original entry on oeis.org

1, 74, 1804128, 53512221536494, 2922627429145967591227933, 497730359833453928180319002991414602093, 403397426941463986598664115278880491308873007636372427413, 2209668743041973325985756217800328983151637526070225333484395817216844313778044

Offset: 1

Marko Riedel, Nov 01 2016

nonn

Scott Burns, Marko Riedel, Markus Scheuer, Number of functions, Math StackExchange.
Scott Burns, Marko Riedel, Markus Scheuer, Enumeration of finite automata, Math StackExchange.
F. Harary and E. Palmer, Enumeration of finite automata, Information and Control, 10 (1967), 499-508.
Marko Riedel, Maple code for A277839-A277844.

Cf. A277839, A277840, A277842, A277843, A277844.

A277842 Number of finite automata with n states, n input symbols and one four output symbols.

Original entry on oeis.org

1, 81, 6064606, 1334647986999812, 970906913413864886205472630, 3914970565374711299589044295533654728633307, 133558404360787903168869516536280931557107488047811301767090944, 54745234941096457415294245370001308972451724232455240696557887565208148810995582605398

Offset: 1

Marko Riedel, Nov 01 2016

nonn

Scott Burns, Marko Riedel, Markus Scheuer, Number of functions, Math StackExchange.
Scott Burns, Marko Riedel, Markus Scheuer, Enumeration of finite automata, Math StackExchange.
F. Harary and E. Palmer, Enumeration of finite automata, Information and Control, 10 (1967), 499-508.
Marko Riedel, Maple code for A277839-A277844.

Cf. A277839, A277840, A277841, A277843, A277844.

A277843 Number of finite automata with n states, n input symbols and five output symbols.

Original entry on oeis.org

1, 81, 9875766, 9508729532667775, 51400728418762283743166947873, 2412787002750586428934439397030434799264061139, 1497241493787657622590696899117249253525915361372369634716838093562, 17442838191172723332310678848004599133452884005515399679140805741625547114446168989072880538

Offset: 1

Marko Riedel, Nov 01 2016

nonn

Scott Burns, Marko Riedel, Markus Scheuer, Number of functions, Math StackExchange.
Scott Burns, Marko Riedel, Markus Scheuer, Enumeration of finite automata, Math StackExchange.
F. Harary and E. Palmer, Enumeration of finite automata, Information and Control, 10 (1967), 499-508.
Marko Riedel, Maple code for A277839-A277844.

Cf. A277839, A277840, A277841, A277842, A277844.

A277836 Number of '6' digits in the set of all numbers from 0 to A014824(n) = Sum_{i=1..n} i*10^(n-i) = (0, 1, 12, 123, 1234, 12345, ...).

Original entry on oeis.org

0, 0, 1, 22, 343, 4664, 58986, 713315, 8367717, 96022849, 1083685281, 12071420713, 133059886145, 1454055651577, 15775124417009, 170096923182441, 1824426021947881, 19478828120713394, 207133960219479637, 2194796392318253180, 23182531824417099723

Offset: 0

M. F. Hasler, Nov 01 2016

			For n=2 there is only one digit '6' in the sequence 0, 1, 2, ..., 12.
For n=3 there are 11 + 10 = 21 more digits '6' in { 16, 26, ..., 56, 60, ..., 69, 76, 86, ..., 116 }, where 66 accounts for two '6's.

David A. Corneth, Table of n, a(n) for n = 0..998

Cf. A277830 - A277838, A277849, A277635, A272525, A083449, A014824.

T[int_Integer, {bndsLow_Integer, bndsUpp_Integer}] := Table[
   Count[
    Flatten[Table[
      IntegerDigits[m],
      {m, 1, Sum[
         10^i - 1,
         {i, n}
         ]/9
       }
      ]],
    int
    ],
   {n, bndsLow, bndsUpp}
   ];
T[6, {0, 7}](* Robert P. P. McKone, Jan 01 2021 *)

print1(c=N=0);for(n=1,8,print1(","c+=sum(k=N+1,N=N*10+n,#select(d->d==6,digits(k)))))

A277836(n,m=6)=if(n>m,A277836(n,m+1)+(m+2)*10^(n-m-1),A277830(n)-(m>n)) \\ M. F. Hasler, Nov 02 2016

a(n) = A277839(n) =  A083449(n) = A277830(n) - 1 for n < 6,
a(n) = A277835(n) - 7*10^(n-6) for n >= 6,
a(n) = A277837(n) + 8*10^(n-7) for n >= 7.

More terms from Lars Blomberg, Nov 05 2016

Removed incorrect b-file. - David A. Corneth, Dec 31 2020

A362897 Array read by antidiagonals: T(n,k) is the number of nonisomorphic multisets of endofunctions on an n-set with k endofunctions.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 7, 7, 1, 1, 1, 13, 74, 19, 1, 1, 1, 22, 638, 1474, 47, 1, 1, 1, 34, 4663, 118949, 41876, 130, 1, 1, 1, 50, 28529, 7643021, 42483668, 1540696, 343, 1, 1, 1, 70, 151600, 396979499, 33179970333, 23524514635, 68343112, 951, 1

Offset: 0

Andrew Howroyd, May 10 2023

Isomorphism is up to permutations of the elements of the n-set.

			Array begins:
======================================================================
n/k| 0   1       2           3               4                   5 ...
---+------------------------------------------------------------------
0  | 1   1       1           1               1                   1 ...
1  | 1   1       1           1               1                   1 ...
2  | 1   3       7          13              22                  34 ...
3  | 1   7      74         638            4663               28529 ...
4  | 1  19    1474      118949         7643021           396979499 ...
5  | 1  47   41876    42483668     33179970333      20762461502595 ...
6  | 1 130 1540696 23524514635 274252613077267 2559276179593762172 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (first 51 antidiagonals).

Columns k=0..3 are A000012, A001372, A054745, A362898.
Row n=2 is A002623.
Main diagonal is A277839.
Cf. A362644.

permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
K(v,m) = {prod(i=1, #v, my(g=gcd(v[i],m), e=v[i]/g); sum(j=1, #v, my(t=v[j]); if(e%(t/gcd(t,m))==0, t))^g)}
T(n,k) = {if(n==0, 1, my(s=0); forpart(q=n, s+=permcount(q) * polcoef(exp(sum(m=1, k, K(q,m)*x^m/m, O(x*x^k))), k)); s/n!)}

T(0,k) = T(1,k) = 1.

A362902 Number of nonisomorphic multisets of fixed-point-free endofunctions on an n-set with n endofunctions.

Original entry on oeis.org

1, 0, 1, 22, 81015, 78954264778, 28097782272713021650, 5303746708038163087962610366540, 750301169575330570948709311656320180808090293, 107340347538849305728376873042317973326061404804760169389400111, 20170881768834535901905153264317820972982675109510814553304001503205327205300015737

Offset: 0

Andrew Howroyd, May 10 2023

nonn

Isomorphism is up to permutation of the elements of the n-set.

Andrew Howroyd, Table of n, a(n) for n = 0..25

Main diagonal of A362899.

Cf. A277839.

cp's OEIS Frontend

A277844 Number of finite automata with n states, n input symbols and six output symbols.

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A277840 Number of finite automata with n states, n input symbols and two output symbols.

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A277841 Number of finite automata with n states, n input symbols and three output symbols.

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A277842 Number of finite automata with n states, n input symbols and one four output symbols.

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A277843 Number of finite automata with n states, n input symbols and five output symbols.

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A277836 Number of '6' digits in the set of all numbers from 0 to A014824(n) = Sum_{i=1..n} i*10^(n-i) = (0, 1, 12, 123, 1234, 12345, ...).

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A362897 Array read by antidiagonals: T(n,k) is the number of nonisomorphic multisets of endofunctions on an n-set with k endofunctions.

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A362902 Number of nonisomorphic multisets of fixed-point-free endofunctions on an n-set with n endofunctions.

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