A027375 -id:A027375 - cp's OEIS frontend

A302291 a(n) is the period of the binary expansion of n.

1, 1, 2, 1, 3, 3, 3, 1, 4, 4, 2, 4, 4, 4, 4, 1, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 1, 6, 6, 6, 6, 3, 6, 6, 6, 6, 6, 2, 6, 6, 3, 6, 6, 6, 6, 6, 6, 6, 6, 3, 6, 6, 6, 6, 6, 6, 6, 6, 1, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7

Offset: 0

Rémy Sigrist, Apr 04 2018

Zero is assumed to be represented as 0; otherwise, leading zeros are ignored.

See A302295 for the variant where leading zeros are allowed.

			The first terms, alongside the binary expansion of n with periodic part in parentheses, are:
  n  a(n)    bin(n)
  -- ----    ------
   0    1    (0)
   1    1    (1)
   2    2    (10)
   3    1    (1)(1)
   4    3    (100)
   5    3    (101)
   6    3    (110)
   7    1    (1)(1)(1)
   8    4    (1000)
   9    4    (1001)
  10    2    (10)(10)
  11    4    (1011)
  12    4    (1100)
  13    4    (1101)
  14    4    (1110)
  15    1    (1)(1)(1)(1)
  16    5    (10000)
  17    5    (10001)
  18    5    (10010)
  19    5    (10011)
  20    5    (10100)

Rémy Sigrist, Table of n, a(n) for n = 0..10000
Index entries for sequences related to binary expansion of n

Aperiodic compositions are counted by A000740.
Aperiodic binary words are counted by A027375.
The orderless period of prime indices is A052409.
Numbers whose binary expansion is periodic are A121016.
Periodic compositions are counted by A178472.
Numbers whose prime signature is aperiodic are A329139.
Compositions by number of distinct rotations are A333941.
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Necklaces are A065609.
- Sum is A070939.
- Runs are counted by A124767.
- Rotational symmetries are counted by A138904.
- Strict compositions are A233564.
- Constant compositions are A272919.
- Lyndon compositions are A275692.
- Co-Lyndon compositions are A326774.
- Aperiodic compositions are A328594.
- Rotational period is A333632.
- Co-necklaces are A333764.
- Reversed necklaces are A333943.
Cf. A000031, A001037, A008965, A019536, A020330, A211100, A302295, A328595, A328596, A329312, A329313, A329326.

Table[If[n==0,1,Length[Union[Array[RotateRight[IntegerDigits[n,2],#]&,IntegerLength[n,2]]]]],{n,0,50}] (* Gus Wiseman, Apr 19 2020 *)

a(n) = my (l=max(1, #binary(n))); fordiv (l, w, if (#Set(digits(n, 2^w))<=1, return (w)))

a(n) = A070939(n) / A138904(n).
a(2^n) = n + 1 for any n >= 0.
a(2^n - 1) = 1 for any n >= 0.
a(A020330(n)) = a(n) for any n > 0.

A323864 Number of aperiodic binary arrays of size n.

Original entry on oeis.org

1, 2, 4, 12, 32, 60, 216, 252, 912, 1494, 3960, 4092, 23904, 16380, 65016, 130920, 324960, 262140, 1569132, 1048572, 6281280, 8388072, 16769016, 16777212, 134150880, 100663050, 268402680, 536865840, 1610449344, 1073741820, 8589664080, 4294967292, 25768888320

Offset: 0

Gus Wiseman, Feb 04 2019

nonn

An n X k matrix is aperiodic if all n * k rotations of its sequence of rows and its sequence of columns are distinct.

			The a(4) = 32 arrays:
  [0001][0010][0011][0100][0110][0111][1000][1001][1011][1100][1101][1110]
.
  [00] [00] [01] [01] [10] [10] [11] [11]
  [01] [10] [00] [11] [00] [11] [01] [10]
.
  [0] [0] [0] [0] [0] [0] [1] [1] [1] [1] [1] [1]
  [0] [0] [0] [1] [1] [1] [0] [0] [0] [1] [1] [1]
  [0] [1] [1] [0] [1] [1] [0] [0] [1] [0] [0] [1]
  [1] [0] [1] [0] [0] [1] [0] [1] [1] [0] [1] [0]

Andrew Howroyd, Table of n, a(n) for n = 0..200

Cf. A000740, A027375, A265627, A323351.

Cf. A323860, A323862, A323863, A323865, A323867, A323869.

apermatQ[m_]:=UnsameQ@@Join@@Table[RotateLeft[m,{i,j}],{i,Length[m]},{j,Length[First[m]]}];
zaz[n_]:=Join@@(Table[Partition[#,d],{d,Divisors[n]}]&/@Tuples[{0,1},n]);
Table[Length[Select[zaz[n],apermatQ]],{n,10}]

a(n) = Sum_{d|n} A323860(d, n/d). - Andrew Howroyd, Aug 21 2019

Terms a(18) and beyond from Andrew Howroyd, Aug 21 2019

A333939 Number of multisets of compositions that can be shuffled together to obtain the k-th composition in standard order.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 2, 3, 1, 2, 2, 4, 2, 5, 4, 5, 1, 2, 2, 4, 2, 4, 5, 7, 2, 5, 4, 10, 4, 10, 7, 7, 1, 2, 2, 4, 2, 5, 5, 7, 2, 5, 3, 9, 5, 13, 11, 12, 2, 5, 5, 10, 5, 11, 13, 18, 4, 10, 9, 20, 7, 18, 12, 11, 1, 2, 2, 4, 2, 5, 5, 7, 2, 4, 4, 11, 5, 14, 11, 12, 2

Offset: 0

Gus Wiseman, Apr 15 2020

nonn

Number of ways to deal out the k-th composition in standard order to form a multiset of hands.

A composition of n is a finite sequence of positive integers summing to n. The k-th composition in standard order (row k of A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The dealings for n = 1, 3, 7, 11, 13, 23, 43:
  (1)  (11)    (111)      (211)      (121)      (2111)        (2211)
       (1)(1)  (1)(11)    (1)(21)    (1)(12)    (11)(21)      (11)(22)
               (1)(1)(1)  (2)(11)    (1)(21)    (1)(211)      (1)(221)
                          (1)(1)(2)  (2)(11)    (2)(111)      (21)(21)
                                     (1)(1)(2)  (1)(1)(21)    (2)(211)
                                                (1)(2)(11)    (1)(1)(22)
                                                (1)(1)(1)(2)  (1)(2)(21)
                                                              (2)(2)(11)
                                                              (1)(1)(2)(2)

Multisets of compositions are counted by A034691.
Combinatory separations of normal multisets are counted by A269134.
Dealings with total sum n are counted by A292884.
Length of co-Lyndon factorization of binary expansion is A329312.
Length of Lyndon factorization of reversed binary expansion is A329313.
All of the following pertain to compositions in standard order (A066099):
- Length is A000120.
- Necklaces are A065609.
- Sum is A070939.
- Runs are counted by A124767.
- Rotational symmetries are counted by A138904.
- Strict compositions are A233564.
- Constant compositions are A272919.
- Lyndon words are A275692.
- Co-Lyndon words are A326774.
- Aperiodic compositions are A328594.
- Length of Lyndon factorization is A329312.
- Distinct rotations are counted by A333632.
- Co-Lyndon factorizations are counted by A333765.
- Lyndon factorizations are counted by A333940.
- Length of co-Lyndon factorization is A334029.
- Combinatory separations are A334030.
Cf. A000031, A000740, A001037, A008965, A027375, A059966, A060223, A211100, A328595, A328596, A333764, A333943.

nn=100;
comps[0]:={{}};comps[n_]:=Join@@Table[Prepend[#,i]&/@comps[n-i],{i,n}];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
dealings[q_]:=Union[Function[ptn,Sort[q[[#]]&/@ptn]]/@sps[Range[Length[q]]]];
Table[Length[dealings[stc[n]]],{n,0,nn}]

For n > 0, Sum_{k = 2^(n-1)..2^n-1} a(k) = A292884(n).

A020921 Triangle read by rows: T(m,n) = number of solutions to 1 <= a(1) < a(2) < ... < a(m) <= n, where gcd( a(1), a(2), ..., a(m), n) = 1.

Original entry on oeis.org

1, 1, 1, 0, 1, 1, 0, 2, 3, 1, 0, 2, 5, 4, 1, 0, 4, 10, 10, 5, 1, 0, 2, 11, 19, 15, 6, 1, 0, 6, 21, 35, 35, 21, 7, 1, 0, 4, 22, 52, 69, 56, 28, 8, 1, 0, 6, 33, 83, 126, 126, 84, 36, 9, 1, 0, 4, 34, 110, 205, 251, 210, 120, 45, 10, 1, 0, 10, 55, 165, 330, 462, 462, 330, 165, 55, 11

Offset: 0

Michael Somos, Nov 17 2002

			From _R. J. Mathar_, Feb 12 2007: (Start)
Triangle begins
  1
  1 1
  0 1  1
  0 2  3   1
  0 2  5   4   1
  0 4 10  10   5   1
  0 2 11  19  15   6   1
  0 6 21  35  35  21   7   1
  0 4 22  52  69  56  28   8  1
  0 6 33  83 126 126  84  36  9  1
  0 4 34 110 205 251 210 120 45 10 1
The inverse of the triangle is
   1
  -1    1
   1   -1    1
  -1    1   -3    1
   1   -1    7   -4    1
  -1    1  -15   10   -5    1
   1   -1   31  -19   15   -6    1
  -1    1  -63   28  -35   21   -7    1
   1   -1  127  -28   71  -56   28   -8    1
  -1    1 -255    1 -135  126  -84   36   -9    1
   1   -1  511   80  255 -251  210 -120   45  -10    1
with row sums 1,0,1,-2,4,-9,22,-55,135,-319,721,...(cf. A038200).
(End)

Temba Shonhiwa, A Generalization of the Euler and Jordan Totient Functions, Fib. Quart., 37 (1999), 67-76.

(Left-hand) columns include A000010, A102309. Row sums are essentially A027375.

Cf. A327029.

A020921 := proc(n,k) option remember ; local divs ; if n <= 0 then 1 ; elif k > n then 0 ; else divs := numtheory[divisors](n) ; add(numtheory[mobius](op(i,divs))*binomial(n/op(i,divs),k),i=1..nops(divs)) ; fi ; end: nmax := 10 ; for row from 0 to nmax do for col from 0 to row do printf("%d,",A020921(row,col)) ; od ; od ; # R. J. Mathar, Feb 12 2007

nmax = 11; t[n_, k_] := Total[ MoebiusMu[#]*Binomial[n/#, k] & /@ Divisors[n]]; t[0, 0] = 1; Flatten[ Table[t[n, k], {n, 0, nmax}, {k, 0, n}]] (* Jean-François Alcover, Oct 20 2011, after PARI *)

{T(n, k) = if( n<=0, k==0 && n==0, sumdiv(n, d, moebius(d) * binomial(n/d, k)))}

# uses[DivisorTriangle from A327029]
DivisorTriangle(moebius, binomial, 13) # Peter Luschny, Aug 24 2019

A329134 Numbers whose differences of prime indices are a periodic word.

Original entry on oeis.org

8, 16, 27, 30, 32, 64, 81, 105, 110, 125, 128, 180, 210, 238, 243, 256, 273, 343, 385, 450, 506, 512, 625, 627, 729, 806, 935, 1001, 1024, 1080, 1100, 1131, 1155, 1331, 1394, 1495, 1575, 1729, 1786, 1870, 1887, 2048, 2187, 2197, 2310, 2401, 2431, 2451, 2635

Offset: 1

Gus Wiseman, Nov 09 2019

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

A sequence is periodic if its cyclic rotations are not all different.

			The sequence of terms together with their differences of prime indices begins:
     8: (0,0)
    16: (0,0,0)
    27: (0,0)
    30: (1,1)
    32: (0,0,0,0)
    64: (0,0,0,0,0)
    81: (0,0,0)
   105: (1,1)
   110: (2,2)
   125: (0,0)
   128: (0,0,0,0,0,0)
   180: (0,1,0,1)
   210: (1,1,1)
   238: (3,3)
   243: (0,0,0,0)
   256: (0,0,0,0,0,0,0)
   273: (2,2)
   343: (0,0)
   385: (1,1)
   450: (1,0,1,0)

Complement of A329135.
These are the Heinz numbers of the partitions counted by A329144.
Periodic binary words are A152061.
Periodic compositions are A178472.
Numbers whose binary expansion is periodic are A121016.
Numbers whose prime signature is periodic are A329140.
Cf. A000740, A027375, A056239, A112798, A124010, A328594, A329132, A329139.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
aperQ[q_]:=Array[RotateRight[q,#1]&,Length[q],1,UnsameQ];
Select[Range[10000],!aperQ[Differences[primeMS[#]]]&]

A329135 Numbers whose differences of prime indices are an aperiodic word.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 28, 29, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73

Offset: 1

Gus Wiseman, Nov 09 2019

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

A sequence is aperiodic if its cyclic rotations are all different.

			The sequence of terms together with their differences of prime indices begins:
    1: ()
    2: ()
    3: ()
    4: (0)
    5: ()
    6: (1)
    7: ()
    9: (0)
   10: (2)
   11: ()
   12: (0,1)
   13: ()
   14: (3)
   15: (1)
   17: ()
   18: (1,0)
   19: ()
   20: (0,2)
   21: (2)
   22: (4)

Complement of A329134.
These are the Heinz numbers of the partitions counted by A329137.
Aperiodic compositions are A000740.
Aperiodic binary words are A027375.
Numbers whose binary expansion is aperiodic are A328594.
Numbers whose prime signature is aperiodic are A329139.
Cf. A056239, A112798, A124010, A152061, A329133, A329136, A329140.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
aperQ[q_]:=Array[RotateRight[q,#1]&,Length[q],1,UnsameQ];
Select[Range[100],aperQ[Differences[primeMS[#]]]&]

A045664 Number of 2n-bead balanced binary strings of fundamental period 2n, rotationally equivalent to reversed complement.

Original entry on oeis.org

1, 2, 4, 18, 48, 150, 324, 882, 1920, 4536, 9900, 22506, 48240, 106470, 227556, 490950, 1044480, 2228190, 4708368, 9961434, 20950800, 44037378, 92229588, 192937938, 402549120, 838860000, 1744617420, 3623864832, 7515733680

Offset: 0

David W. Wilson

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Cf. A000740, A001037, A007727, A027375, A045662, A045663.

a[n_] := If[n == 0, 1, 2n Sum[MoebiusMu[n/d] 2^(d-1), {d, Divisors[n]}]];
a /@ Range[0, 30] (* Jean-François Alcover, Sep 23 2019, from PARI *)

a(n)={if(n<1, n==0, 2*n*sumdiv(n, d, moebius(n/d)*2^(d-1)))} \\ Andrew Howroyd, Sep 14 2019

For n >= 1, a(n) = n*A027375(n) = 2*n*A000740(n) = n^2*A001037(n).

a(n) = 2*n*Sum_{d|n} mu(n/d)*2^(d-1) for n > 0. - Andrew Howroyd, Sep 14 2019

A059412 Number of distinct minimal unary DFA's with exactly n states.

Original entry on oeis.org

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

Offset: 1

Jeffrey Shallit, Jan 30 2001

nonn

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

			a(1) = 2 because there are exactly two minimal unary automata with 1 state.

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.

Cyril Nicaud, Average state complexity of operations on unary automata, Math. Foundations of Computer Science, 1999, Lect. Notes in Computer Sci. #1672, pp. 231-240.
Lucas B. Vieira and Costantino Budroni, Temporal correlations in the simplest measurement sequences, Quantum 6 p. 623 (2022).
Lucas Vieira Barbosa, Nonclassical Temporal Correlations Under Finite-Memory Constraints, Doctoral Thesis, Univ. Wien (Austria 2024). See p. 35.

A059412 := proc(n)
    A027375(n)+add(A027375(n-j)*2^(j-1),j=1..n-1) ;
end proc:
seq(A059412(n),n=1..10) ; # R. J. Mathar, May 21 2018

A027375[n_] := DivisorSum[n, MoebiusMu[n/#]*2^# &];
a[n_] := A027375[n] + Sum[A027375[n-j]*2^(j-1), {j, 1, n-1}];
Table[a[n], {n, 1, 29}] (* Jean-François Alcover, May 08 2023 *)

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

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).

a(n) = A027375(n) + A059413(n-1). - R. J. Mathar, May 21 2018

A323863 Number of n X n aperiodic binary arrays.

Original entry on oeis.org

1, 2, 8, 486, 64800, 33554250, 68718675672, 562949953420302, 18446744060824780800, 2417851639229257812542976, 1267650600228226023797043513000, 2658455991569831745807614120560664598, 22300745198530623141521551172073990303938400

Offset: 0

Gus Wiseman, Feb 04 2019

nonn

An n X k matrix is aperiodic if all n * k rotations of its sequence of rows and its sequence of columns are distinct.

			The a(2) = 8 arrays are:
  [0 0] [0 0] [0 1] [0 1] [1 0] [1 0] [1 1] [1 1]
  [0 1] [1 0] [0 0] [1 1] [0 0] [1 1] [0 1] [1 0]
Note that the following are not aperiodic even though their row and column sequences are (independently) aperiodic:
  [1 0] [0 1]
  [0 1] [1 0]

Andrew Howroyd, Table of n, a(n) for n = 0..50

Cf. A000031, A000740, A027375, A179043, A265627, A323351.

Cf. A323860, A323862, A323864, A323865, A323867, A323869.

apermatQ[m_]:=UnsameQ@@Join@@Table[RotateLeft[m,{i,j}],{i,Length[m]},{j,Length[First[m]]}];
Table[Length[Select[(Partition[#,n]&)/@Tuples[{0,1},n^2],apermatQ]],{n,4}]

a(n) = 2^(n^2) - (n+1)*2^n + 2*n if n is prime. - Robert Israel, Feb 04 2019

a(n) = n^2 * A323872(n). - Andrew Howroyd, Aug 21 2019

a(5) from Robert Israel, Feb 04 2019
a(6)-a(7) from Giovanni Resta, Feb 05 2019
Terms a(8) and beyond from Andrew Howroyd, Aug 21 2019

A324462 Number of simple graphs covering n vertices with distinct rotations.

Original entry on oeis.org

1, 0, 0, 3, 28, 765, 26958, 1887277, 252458904, 66376420155, 34508978662350, 35645504882731557, 73356937843604425644, 301275024444053951967585, 2471655539736990372520379226, 40527712706903544100966076156895, 1328579255614092949957261201822704816

Offset: 0

Gus Wiseman, Feb 28 2019

nonn

A simple graph with n vertices has distinct rotations if all n rotations of its vertex set act on the edge set to give distinct graphs. It is covering if there are no isolated vertices. These are different from aperiodic graphs and acyclic graphs but are similar to aperiodic sequences (A000740) and aperiodic arrays (A323867).

Andrew Howroyd, Table of n, a(n) for n = 0..50
Gus Wiseman, The a(4) = 28 graph covers with distinct rotations.
Gus Wiseman, The a(4) = 28 graph covers with distinct rotations, radial embedding.

Cf. A000088, A000740, A002494, A006125, A006129, A027375, A192332, A323863, A323864, A323867, A323869, A324461 (non-covering case), A324463, A324464.

rotgra[g_,m_]:=Sort[Sort/@(g/.k_Integer:>If[k==m,1,k+1])];
Table[Length[Select[Subsets[Subsets[Range[n],{2}]],And[Union@@#==Range[n],UnsameQ@@Table[Nest[rotgra[#,n]&,#,j],{j,n}]]&]],{n,0,5}]

a(n)={if(n<1, n==0, sumdiv(n, d, moebius(n/d)*sum(k=0, d, (-1)^(d-k)*binomial(d,k)*2^(k*(k-1)*n/(2*d) + k*(n/d\2)))))} \\ Andrew Howroyd, Aug 19 2019

a(n) = Sum{d|n} mu(n/d) * Sum_{k=0..d} (-1)^(d-k)*binomial(d,k)*2^( k*(k-1)*n/(2*d) + k*(floor(n/(2*d))) ) for n > 0. - Andrew Howroyd, Aug 19 2019

Terms a(7) and beyond from Andrew Howroyd, Aug 19 2019

cp's OEIS Frontend

A302291 a(n) is the period of the binary expansion of n.

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A323864 Number of aperiodic binary arrays of size n.

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A333939 Number of multisets of compositions that can be shuffled together to obtain the k-th composition in standard order.

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A020921 Triangle read by rows: T(m,n) = number of solutions to 1 <= a(1) < a(2) < ... < a(m) <= n, where gcd( a(1), a(2), ..., a(m), n) = 1.

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A329134 Numbers whose differences of prime indices are a periodic word.

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A329135 Numbers whose differences of prime indices are an aperiodic word.

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Mathematica

A045664 Number of 2n-bead balanced binary strings of fundamental period 2n, rotationally equivalent to reversed complement.

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PARI

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A059412 Number of distinct minimal unary DFA's with exactly n states.

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