A000048 -id:A000048 - cp's OEIS frontend

A000016 a(n) is the number of distinct (infinite) output sequences from binary n-stage shift register which feeds back the complement of the last stage.

1, 1, 1, 2, 2, 4, 6, 10, 16, 30, 52, 94, 172, 316, 586, 1096, 2048, 3856, 7286, 13798, 26216, 49940, 95326, 182362, 349536, 671092, 1290556, 2485534, 4793492, 9256396, 17895736, 34636834, 67108864, 130150588, 252645136, 490853416

Offset: 0

N. J. A. Sloane

Also a(n+1) = number of distinct (infinite) output sequences from binary n-stage shift register which feeds back the complement of the sum of its contents. E.g., for n=5 there are 6 such sequences.
Also a(n+1) = number of binary vectors (x_1,...x_n) satisfying Sum_{i=1..n} i*x_i = 0 (mod n+1) = size of Varshamov-Tenengolts code VT_0(n). E.g., |VT_0(5)| = 6 = a(6).
Number of binary necklaces with an odd number of zeros. - Joerg Arndt, Oct 26 2015
Also, number of subsets of {1,2,...,n-1} which sum to 0 modulo n (cf. A063776). - Max Alekseyev, Mar 26 2016
From Gus Wiseman, Sep 14 2019: (Start)
Also the number of subsets of {1..n} containing n whose mean is an element. For example, the a(1) = 1 through a(8) = 16 subsets are:
  1  2  3    4    5      6      7        8
        123  234  135    246    147      258
                  345    456    357      468
                  12345  1236   567      678
                         1456   2347     1348
                         23456  2567     1568
                                12467    3458
                                13457    3678
                                34567    12458
                                1234567  14578
                                         23578
                                         24568
                                         45678
                                         123468
                                         135678
                                         2345678
(End)
Number of self-dual binary necklaces with 2n beads (cf. A263768, A007147). - Bernd Mulansky, Apr 25 2023

			For n=3 the 2 output sequences are 000111000111... and 010101...
For n=5 the 4 output sequences are those with periodic parts {0000011111, 0001011101, 0010011011, 01}.
For n=6 there are 6 such sequences.

B. D. Ginsburg, On a number theory function applicable in coding theory, Problemy Kibernetiki, No. 19 (1967), pp. 249-252.
S. W. Golomb, Shift-Register Sequences, Holden-Day, San Francisco, 1967, p. 172.
J. Hedetniemi and K. R. Hutson, Equilibrium of shortest path load in ring network, Congressus Numerant., 203 (2010), 75-95. See p. 83.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane, On single-deletion-correcting codes, in Codes and Designs (Columbus, OH, 2000), 273-291, Ohio State Univ. Math. Res. Inst. Publ., 10, de Gruyter, Berlin, 2002.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
D. Stoffer, Delay equations with rapidly oscillating stable periodic solutions, J. Dyn. Diff. Eqs. 20 (1) (2008) 201, eq. (39)

Seiichi Manyama, Table of n, a(n) for n = 0..3334 (first 201 terms from T. D. Noe)
Nicolás Álvarez, Victória Becher, Martín Mereb, Ivo Pajor, and Carlos Miguel Soto, On extremal factors of de Bruijn-like graphs, Univ. Buenos Aires (Argentina 2023). See also arXiv:2308.16257 [math.CO], 2023. See references.
Joshua P. Bowman, Compositions with an Odd Number of Parts, and Other Congruences, J. Int. Seq (2024) Vol. 27, Art. 24.3.6. See p. 17.
A. E. Brouwer, The Enumeration of Locally Transitive Tournaments, Math. Centr. Report ZW138, Amsterdam, 1980.
S. Butenko, P. Pardalos, I. Sergienko, V. P. Shylo and P. Stetsyuk, Estimating the size of correcting codes using extremal graph problems, Optimization, 227-243, Springer Optim. Appl., 32, Springer, New York, 2009.
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Sébastien Designolle, Tamás Vértesi, and Sebastian Pokutta, Symmetric multipartite Bell inequalities via Frank-Wolfe algorithms, arXiv:2310.20677 [quant-ph], 2023.
T. M. A. Fink, Exact dynamics of the critical Kauffman model with connectivity one, arXiv:2302.05314 [cond-mat.stat-mech], 2023.
R. W. Hall and P. Klingsberg, Asymmetric rhythms and tiling canons, Amer. Math. Monthly, 113 (2006), 887-896.
A. A. Kulkarni, N. Kiyavash and R. Sreenivas, On the Varshamov-Tenengolts Construction on Binary Strings, 2013.
E. M. Palmer and R. W. Robinson, Enumeration of self-dual configurations, Pacific J. Math., 110 (1984), 203-221.
R. Pries and C. Weir, The Ekedahl-Oort type of Jacobians of Hermitian curves, arXiv preprint arXiv:1302.6261 [math.NT], 2013.
N. J. A. Sloane, On single-deletion-correcting codes
N. J. A. Sloane, Challenge Problems: Independent Sets in Graphs
Yan Bo Ti, Gabriel Verret, and Lukas Zobernig, Abelian Varieties with p-rank Zero, arXiv:2203.08401 [math.NT], 2022.
Antonio Vera López, Luis Martínez, Antonio Vera Pérez, Beatriz Vera Pérez, and Olga Basova, Combinatorics related to Higman's conjecture I: Parallelogramic digraphs and dispositions, Linear Algebra Appl. 530, 414-444 (2017).
Index entries for sequences related to tournaments
Index entries for sequences related to necklaces
Index entries for sequences related to subset sums modulo m

The main diagonal of table A068009, the left edge of triangle A053633.
Cf. A000048, A000031, A000013, A053634, A182469.
Subsets whose mean is an element are A065795.
Dominated by A082550.
Partitions containing their mean are A237984.
Subsets containing n but not their mean are A327477.
Cf. A051293, A135342, A240850, A327471, A327478.

a000016 0 = 1
a000016 n = (`div` (2 * n)) $ sum $
   zipWith (*) (map a000010 oddDivs) (map ((2 ^) . (div n)) $ oddDivs)
   where oddDivs = a182469_row n
-- Reinhard Zumkeller, May 01 2012

A000016 := proc(n) local d, t; if n = 0 then return 1 else t := 0; for d from 1 to n do if n mod d = 0 and d mod 2 = 1 then t := t + NumberTheory:-Totient(d)* 2^(n/d)/(2*n) fi od; return t fi end:

a[0] = 1; a[n_] := Sum[Mod[k, 2] EulerPhi[k]*2^(n/k)/(2*n), {k, Divisors[n]}]; Table[a[n], {n, 0, 35}](* Jean-François Alcover, Feb 17 2012, after Pari *)

a(n)=if(n<1,n >= 0,sumdiv(n,k,(k%2)*eulerphi(k)*2^(n/k))/(2*n));

from sympy import totient, divisors
def A000016(n): return sum(totient(d)<>(~n&n-1).bit_length(),generator=True))//n if n else 1 # Chai Wah Wu, Feb 21 2023

a(n) = Sum_{odd d divides n} (phi(d)*2^(n/d))/(2*n), n>0.
a(n) = A063776(n)/2.
a(n) = 2^(n-1) - A327477(n). - Gus Wiseman, Sep 14 2019

More terms from Michael Somos, Dec 11 1999

A000013 Definition (1): Number of n-bead binary necklaces with beads of 2 colors where the colors may be swapped but turning over is not allowed.

Original entry on oeis.org

1, 1, 2, 2, 4, 4, 8, 10, 20, 30, 56, 94, 180, 316, 596, 1096, 2068, 3856, 7316, 13798, 26272, 49940, 95420, 182362, 349716, 671092, 1290872, 2485534, 4794088, 9256396, 17896832, 34636834, 67110932, 130150588, 252648992, 490853416, 954444608, 1857283156, 3616828364

Offset: 0

N. J. A. Sloane

Definition (2): Equivalently, number of different output sequences from an n-stage pure cycling shift register when 2 sequences are considered the same if one is the complement of the other.
Definition (3): Also number of different output sequences from an n-stage pure cycling shift register constrained so contents have even weight.
Definition (4): Also number of output sequences from (n-1)-stage shift register which feeds back the mod 2 sum of the contents of the register.
The equivalence of definitions (1) and (2) follows at once from the definitions.
If u is an output sequence of type (2) then its derivative is of type (3) - so (2) and (3) count the same things.
If we have a shift register of type (4), append a new cell which contains the mod 2 sum of the contents to get a shift register of type (3). So (3) and (4) count the same things.
If n is even, a(n) = A000116(n/2). If 2^(n+1)-1 is prime, then a(n) = A128976(n+1), the number of cycles in the digraph of the Lucas-Lehmer operator LL(x) = x^2 - 2 acting on Z/(2^(n+1)-1). - M. F. Hasler, May 19 2007
Also number of 2n-bead balanced binary necklaces that are equivalent to their complements. - Andrew Howroyd, Sep 29 2017

			G.f. = 1 + x + 2*x^2 + 2*x^3 + 4*x^4 + 4*x^5 + 8*x^6 + 10*x^7 + 20*x^8 + ...

S. W. Golomb, Shift-Register Sequences, Holden-Day, San Francisco, 1967, p. 172.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 0..3334 (first 201 terms from T. D. Noe)
Nicolás Álvarez, Victória Becher, Martín Mereb, Ivo Pajor, and Carlos Miguel Soto, On extremal factors of de Bruijn-like graphs, Univ. Buenos Aires (Argentina 2023). See also arXiv:2308.16257 [math.CO], 2023. See references.
Joerg Arndt, Matters Computational (The Fxtbook), p. 151, p. 408.
Raghav Bhutani and Frederick Saia, Replacement dynamics of binary quadratic forms, arXiv:2508.05816 [math.NT], 2025. See p. 6.
Henry Bottomley, Initial terms of A000011 and A000013
Zachary E. Chin and Isaac L. Chuang, The quantum trajectory sensing problem and its solution, arXiv:2410.00893 [quant-ph], 2024. See p. 19.
N. J. Fine, Classes of periodic sequences, Illinois J. Math., 2 (1958), 285-302.
E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.
Darij Grinberg and Peter Mao, Necklaces over a group with identity product, arXiv:2405.08937 [math.CO], 2024. See pp. 15, 22.
Karyn McLellan, Periodic coefficients and random Fibonacci sequences, Electronic Journal of Combinatorics, 20(4), 2013, #P32.
Frank Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc.
Frank Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc. [Cached copy, with permission, pdf format only]
N. J. A. Sloane, On single-deletion-correcting codes
N. J. A. Sloane, On single-deletion-correcting codes, arXiv:math/0207197 [math.CO], 2002; in Codes and Designs (Columbus, OH, 2000), 273-291, Ohio State Univ. Math. Res. Inst. Publ., 10, de Gruyter, Berlin, 2002.
N. J. A. Sloane, Maple code for this and related sequences
Index entries for sequences related to necklaces

Cf. A000031, A000016, A000116.
Cf. A128976.
Cf. A000010, A027750.
Cf. A385665, A000048.

a000013 0 = 1
a000013 n = sum (zipWith (*)
   (map (a000010 . (* 2)) ds) (map (2 ^) $ reverse ds)) `div` (2 * n)
   where ds = a027750_row n
-- Reinhard Zumkeller, Jul 08 2013

with(numtheory): A000013 := proc(n) local s,d; if n = 0 then RETURN(1) else s := 0; for d in divisors(n) do s := s+(phi(2*d)*2^(n/d))/(2*n); od; RETURN(s); fi; end;

a[n_] := Fold[ #1 + EulerPhi[2#2]2^(n/#2)/(2n) &, 0, Divisors[n]]
a[ n_] := If[ n < 1, Boole[n == 0], DivisorSum[ n, EulerPhi[2 #] 2^(n/#) &] / (2 n)]; (* Michael Somos, Dec 19 2014 *)
mx=40;CoefficientList[Series[1-Sum[EulerPhi[2i] Log[1-2*x^i]/(2i),{i,1,mx}],{x,0,mx}],x] (* Herbert Kociemba, Nov 01 2016 *)

{a(n) = if( n<1, n==0, sumdiv(n, k, eulerphi(2*k) * 2^(n/k)) / (2*n))}; /* Michael Somos, Oct 20 1999 */

from sympy import divisors, totient
def a(n): return 1 if n<1 else sum([totient(2*d)*2**(n//d) for d in divisors(n)])//(2*n) # Indranil Ghosh, Apr 28 2017

a(n) = Sum_{ d divides n } (phi(2*d)*2^(n/d))/(2*n) for n>0. - Michael Somos, Oct 20 1999
G.f.: 1 - Sum_{i>=1} phi(2*i)*log(1-2*x^i)/(2*i). - Herbert Kociemba, Nov 01 2016
From Richard L. Ollerton, May 11 2021: (Start)
For n >= 1:
a(n) = (1/(2*n))*Sum_{k=1..n} phi(2*gcd(n,k))*2^(n/gcd(n,k))/phi(n/gcd(n,k)), where phi = A000010.
a(n) = (1/(2*n))*Sum_{k=1..n} phi(2*n/gcd(n,k))*2^gcd(n,k)/phi(n/gcd(n,k)). (End)
a(n) ~ 2^(n-1)/n. - Cedric Lorand, Apr 24 2022
a(n) = Sum_{k=1..n} A385665(n,k) = Sum_{d|n} A000048(d). - Tilman Piesk, Jul 31 2025

A001372 Number of unlabeled mappings (or mapping patterns) from n points to themselves; number of unlabeled endofunctions.

Original entry on oeis.org

1, 1, 3, 7, 19, 47, 130, 343, 951, 2615, 7318, 20491, 57903, 163898, 466199, 1328993, 3799624, 10884049, 31241170, 89814958, 258604642, 745568756, 2152118306, 6218869389, 17988233052, 52078309200, 150899223268, 437571896993, 1269755237948, 3687025544605, 10712682919341, 31143566495273, 90587953109272, 263627037547365

Offset: 0

N. J. A. Sloane

			The a(3) = 7 mappings are:
1->1, 2->2, 3->3
1->1, 2->2, 3->1 (equiv. to 1->1, 2->2, 3->2, or 1->1, 2->1, 3->3, etc.)
1->1, 2->3, 3->2
1->1, 2->1, 3->2
1->1, 2->1, 3->1
1->2, 2->3, 3->1
1->2, 2->1, 3->1

F. Bergeron, G. Labelle, and P. Leroux, Combinatorial Species and Tree-Like Structures, Cambridge, 1998, pp. 41, 209.
S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 5.6.6.
R. A. Fisher, Contributions to Mathematical Statistics, Wiley, 1950, 41.401.
F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 70, Table 3.4.1.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..1000 (first 501 terms from Christian G. Bower)
A. L. Agore, A. Chirvasitu, and G. Militaru, The set-theoretic Yang-Baxter equation, Kimura semigroups and functional graphs, arXiv:2303.06700 [math.QA], 2023.
N. G. de Bruijn, Enumeration of mapping patterns, J. Combin. Theory, 12 (1972), 14-20.
N. G. de Bruijn and D. A. Klarner, Multisets of aperiodic cycles, SIAM J. Algebraic Discrete Methods 3 (1982), no. 3, 359--368. MR0666861(84i:05008).
Robert L. Davis, The number of structures of finite relations, Proc. Amer. Math. Soc. 4 (1953), 486-495.
Oscar Defrain, Antonio E. Porreca and Ekaterina Timofeeva, Polynomial-delay generation of functional digraphs up to isomorphism, arXiv:2302.13832 [cs.DS], 2023-2024.
Oscar Defrain, Antonio E. Porreca and Ekaterina Timofeeva, Polynomial-delay generation of functional digraphs up to isomorphism, Disc. Appl. Math., vol 357 (2024), pp. 24-33.
Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics, 2009; see page 480
Alexander Heaton, Songpon Sriwongsa, and Jeb F. Willenbring, Branching from the General Linear Group to the Symmetric Group and the Principal Embedding, Algebraic Combinatorics, vol. 4, issue 2 (2021), p. 189-200.
Florent Hivert, Jean-Christophe Novelli, and Jean-Yves Thibon, Commutative combinatorial Hopf algebras, arXiv:math/0605262 [math.CO], 2006.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 144
Sujoy Mukherjee and Petr Vojtěchovský, Minimal representatives of endofunctions, Univ. Denver (2024).
Ronald C. Read, Note on number of functional digraphs, Math. Ann., vol. 143 (1961), pp. 109-111.
Marko Riedel, stackexchange.com, Enumeration of functions by Simultaneous Power Group Enumeration (SPGE)
Marko Riedel, Maple code for sequence using SPGE
Sara Riva, Factorisation of Discrete Dynamical Systems, Ph.D. Thesis, Univ. Côte d'Azur (France 2023).
N. J. A. Sloane, Illustration of initial terms
N. J. A. Sloane, Transforms
P. R. Stein, Letter to N. J. A. Sloane, Jun 02 1971

Cf. A000312, A002861, A006961, A001373, A054050, A054745.

with(combstruct): M[ 2671 ] := [ F,{F=Set(K), K=Cycle(T), T=Prod(Z,Set(T))},unlabeled ]:
a:=seq(count(M[2671],size=n),n=0..27); # added by W. Edwin Clark, Nov 23 2010

Needs["Combinatorica`"];
nn=30;s[n_,k_]:=s[n,k]=a[n+1-k]+If[n<2 k,0,s[n-k,k]];a[1]=1;a[n_]:=a[n]=Sum[a[i] s[n-1,i] i,{i,1,n-1}]/(n-1);rt=Table[a[i],{i,1,nn}];c=Drop[Apply[Plus,Table[Take[CoefficientList[CycleIndex[CyclicGroup[n],s]/.Table[s[j]->Table[Sum[rt[[i]] x^(k*i),{i,1,nn}],{k,1,nn}][[j]],{j,1,nn}],x],nn],{n,1,30}]],1];CoefficientList[Series[Product[1/(1-x^i)^c[[i]],{i,1,nn-1}],{x,0,nn}],x]  (* after code given by Robert A. Russell in A000081 *) (* Geoffrey Critzer, Oct 12 2012 *)
max = 40; A[n_] := A[n] = If[n <= 1, n, Sum[DivisorSum[j, #*A[#]&]*A[n-j], {j, 1, n-1}]/(n-1)]; H[t_] := Sum[A[n]*t^n, {n, 0, max}]; F = 1 / Product[1 - H[x^n], {n, 1, max}] + O[x]^max; CoefficientList[F, x] (* Jean-François Alcover, Dec 01 2015, after Joerg Arndt *)

N=66;  A=vector(N+1, j, 1);
for (n=1, N, A[n+1] = 1/n * sum(k=1, n, sumdiv(k, d, d * A[d]) * A[n-k+1] ) );
A000081=concat([0], A);
H(t)=subst(Ser(A000081, 't), 't, t);
x='x+O('x^N);
F=1/prod(n=1,N, 1 - H(x^n));
Vec(F)
\\ Joerg Arndt, Jul 10 2014

Euler transform of A002861.

a(n) ~ c * d^n / sqrt(n), where d = A051491 = 2.9557652856519949747148... (Otter's rooted tree constant), c = 0.442876769782206479836... (for a closed form see "Mathematical Constants", p.308). - Vaclav Kotesovec, Mar 17 2015

More terms etc. from Paul Zimmermann, Mar 15 1996

Name edited by Keith J. Bauer, Jan 07 2024

A051841 Number of binary Lyndon words with an even number of 1's.

Original entry on oeis.org

1, 0, 1, 1, 3, 4, 9, 14, 28, 48, 93, 165, 315, 576, 1091, 2032, 3855, 7252, 13797, 26163, 49929, 95232, 182361, 349350, 671088, 1290240, 2485504, 4792905, 9256395, 17894588, 34636833, 67106816, 130150493, 252641280, 490853403, 954429840, 1857283155, 3616800768, 7048151355, 13743869130, 26817356775

Offset: 1

Frank Ruskey, Dec 13 1999

Also number of trace 0 irreducible polynomials over GF(2).

Also number of trace 0 Lyndon words over GF(2).

			a(5) = 3 = |{ 00011, 00101, 01111 }|.

May, Robert M. "Simple mathematical models with very complicated dynamics." Nature, Vol. 261, June 10, 1976, pp. 459-467; reprinted in The Theory of Chaotic Attractors, pp. 85-93. Springer, New York, NY, 2004. The sequences listed in Table 2 are A000079, A027375, A000031, A001037, A000048, A051841. - N. J. A. Sloane, Mar 17 2019

T. D. Noe, Table of n, a(n) for n = 1..300
F. Ruskey, Number of q-ary Lyndon words with given trace mod q
F. Ruskey, Number of Lyndon words over GF(q) with given trace
Index entries for sequences related to Lyndon words

Same as A001037 - A000048. Same as A042980 + A042979.
Cf. A027750, A008683.
Cf. A000010.

a051841 n = (sum $ zipWith (\u v -> gcd 2 u * a008683 u * 2 ^ v)
             ds $ reverse ds) `div` (2 * n) where ds = a027750_row n
-- Reinhard Zumkeller, Mar 17 2013

a[n_] := Sum[GCD[d, 2]*MoebiusMu[d]*2^(n/d), {d, Divisors[n]}]/(2n);
Table[a[n], {n, 1, 32}]
(* Jean-François Alcover, May 14 2012, from formula *)

L(n, k) = sumdiv(gcd(n,k), d, moebius(d) * binomial(n/d, k/d) );
a(n) = sum(k=0, n, if( (n+k)%2==0, L(n, k), 0 ) ) / n;
vector(33,n,a(n))
/* Joerg Arndt, Jun 28 2012 */

a(n) = 1/(2*n)*Sum_{d|n} gcd(d,2)*mu(d)*2^(n/d).
a(n) ~ 2^(n-1) / n. - Vaclav Kotesovec, May 31 2019
From Richard L. Ollerton, May 10 2021: (Start)
a(n) = 1/(2*n)*Sum_{k=1..n} gcd(gcd(n,k),2)*mu(gcd(n,k))*2^(n/gcd(n,k))/phi(n/gcd(n,k)).
a(n) = (1/n)*Sum_{k=1..n} gcd(n/gcd(n,k),2)*mu(n/gcd(n,k))*2^gcd(n,k)/phi(n/gcd(n,k)). (End)

A054660 Number of monic irreducible polynomials over GF(4) of degree n with fixed nonzero trace.

Original entry on oeis.org

1, 2, 5, 16, 51, 170, 585, 2048, 7280, 26214, 95325, 349520, 1290555, 4793490, 17895679, 67108864, 252645135, 954437120, 3616814565, 13743895344, 52357696365, 199911205050, 764877654105, 2932031006720, 11258999068416

Offset: 1

N. J. A. Sloane, Apr 18 2000

Also number of Lyndon words of length n with trace 1 over GF(4).
Let x = RootOf( z^2+z+1 ) and y = 1+x. Also number of Lyndon words of length n with trace x over GF(4). Also number of Lyndon words of length n with trace y over GF(4).
Also number of 4-ary Lyndon words (i.e., Lyndon words over Z_4) of length n with trace 1 (mod 4). Also the same with trace 3 (mod 4). - Andrey Zabolotskiy, Dec 19 2020

			a(3; y)=5 since the five Lyndon words over GF(4) of trace y and length 3 are { 00y, 01x, 0x1, 11y, xxy }; the five Lyndon words over Z_4 of trace 1 (mod 4) and length 3 are { 001, 023, 032, 113, 122 }.

Seiichi Manyama, Table of n, a(n) for n = 1..1000
F. Ruskey, Number of monic irreducible polynomials over GF(q) with given trace
F. Ruskey, Number of q-ary Lyndon words with given trace mod q
F. Ruskey, Number of Lyndon words over GF(q) with given trace

Cf. A000048, A051841, A046211, A046209, A054661, etc.
Cf. A008683, A054661, A027377, A300628, A074033, A074034, A074035, A074448, A074449, A074450.
Cf. A074406, A074407, A074408, A074409.

From Seiichi Manyama, Mar 11 2018: (Start)
a(n) = A000048(2*n) = (1/(4*n)) * Sum_{odd d divides n} mu(d)*4^(n/d), where mu is the Möbius function A008683.
a(n+1) = A300628(n,n) for n >= 0. (End)
From Andrey Zabolotskiy, Dec 19 2020: (Start)
a(n) = A074033(n) + A074034(n) + 2 * A074035(n).
a(n) = A074448(n) + A074449(n) + 2 * A074450(n).
a(n) = A074406(n) + A074407(n) + A074408(n) + A074409(n). (End)

More terms from James Sellers, Apr 19 2000

A000046 Number of primitive n-bead necklaces (turning over is allowed) where complements are equivalent.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 5, 8, 14, 21, 39, 62, 112, 189, 352, 607, 1144, 2055, 3885, 7154, 13602, 25472, 48670, 92204, 176770, 337590, 649341, 1246840, 2404872, 4636389, 8964143, 17334800, 33587072, 65107998, 126387975, 245492232, 477349348, 928772649, 1808669170, 3524337789, 6872471442

Offset: 0

N. J. A. Sloane

Also, number of "twills" (Grünbaum and Shephard). - N. J. A. Sloane, Oct 21 2015

			For a(7)=8, there are seven achiral set partitions (0000001, 0000011, 0000101, 0000111, 0001001, 0010011, 0010101) and one chiral pair (0001011-0001101). - _Robert A. Russell_, Jun 19 2019

B. Grünbaum and G. C. Shephard, The geometry of fabrics, pp. 77-98 of F. C. Holroyd and R. J. Wilson, editors, Geometrical Combinatorics. Pitman, Boston, 1984.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Christian G. Bower, Table of n, a(n) for n = 0..1000
E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.
J. E. Iglesias, A formula for the number of closest packings of equal spheres having a given repeat period, Z. Krist. 155 (1981) 121-127, Table 1.
Sara Jensen, Sequence knitting, J. Math. Arts (2023).
Karyn McLellan, Periodic coefficients and random Fibonacci sequences, Electronic Journal of Combinatorics, 20(4), 2013, #P32.
Index entries for sequences related to necklaces

Similar to A000011, but counts primitive necklaces.
A000048 (oriented), A308706 (chiral), A179781 (achiral).
Cf. A054199.

with(numtheory); A000046 := proc(n) local s,d; if n = 0 then RETURN(1); else s := 0; for d in divisors(n) do s := s+mobius(d)*A000011(n/d); od; RETURN(s); fi; end;

a11[0] = 1; a11[n_] := 2^Floor[n/2]/2 + Sum[EulerPhi[2*d]*2^(n/d), {d, Divisors[n]}]/n/4; a[0] = 1; a[n_] := Sum[MoebiusMu[d]*a11[n/d], {d, Divisors[n]}]; Table[a[n], {n, 0, 36}] (* Jean-François Alcover, Jul 10 2012, from formula *)
Join[{1}, Table[(DivisorSum[NestWhile[#/2 &, n, EvenQ], MoebiusMu[#] 2^(n/#) &]/(2 n) + DivisorSum[n, MoebiusMu[n/#] 2^Floor[#/2] &])/2, {n, 1, 40}]] (* Robert A. Russell, Jun 19 2019 *)

apply( {A000046(n)=if(n, sumdiv(n, d, moebius(d)*A000011(n/d)), 1)}, [0..40]) \\ M. F. Hasler, May 27 2025

a(n) = Sum_{ d divides n } mu(d)*A000011(n/d).
From Robert A. Russell, Jun 19 2019: (Start)
a(n) = ((1/(2n))Sum_{odd d|n} mu(d)*2^(n/d) + Sum_{d|n} mu(n/d)*2^floor(d/2)) / 2.
a(n) = A000048(n) - A308706(n) = (A000048(n) + A179781(n))/2 = A308706(n) + A179781(n).
A000011(n) =  Sum_{d|n} a(d). (End)

A325012 Array read by descending antidiagonals: A(n,k) is the number of oriented colorings of the facets of a regular n-dimensional orthoplex using up to k colors.

Original entry on oeis.org

1, 4, 1, 9, 6, 1, 16, 24, 23, 1, 25, 70, 333, 496, 1, 36, 165, 2916, 230076, 2275974, 1, 49, 336, 16725, 22456756, 965227578201, 800648638402240, 1, 64, 616, 70911, 795467350, 9607713956430560, 149031415906337877339236058, 1054942853799126580390222487977120, 1

Offset: 1

Robert A. Russell, May 27 2019

Also called cross polytope and hyperoctahedron. For n=1, the figure is a line segment with two vertices. For n=2 the figure is a square with four edges. For n=3 the figure is an octahedron with eight triangular faces. For n=4, the figure is a 16-cell with sixteen tetrahedral facets. The Schläfli symbol, {3,...,3,4}, of the regular n-dimensional orthoplex (n>1) consists of n-2 threes followed by a four. Each of its 2^n facets is an (n-1)-dimensional simplex. Two oriented colorings are the same if one is a rotation of the other; chiral pairs are counted as two.

Also the number of oriented colorings of the vertices of a regular n-dimensional orthotope (cube) using up to k colors.

			Array begins with A(1,1):
1   4      9       16        25          36           49            64 ...
1   6     24       70       165         336          616          1044 ...
1  23    333     2916     16725       70911       241913        701968 ...
1 496 230076 22456756 795467350 14697611496 173107727191 1466088119056 ...
For A(1,2) = 4, the two achiral colorings use just one of the two colors for both vertices; the chiral pair uses one color for each vertex.

Robert A. Russell, Table of n, a(n) for n = 1..78
E. M. Palmer and R. W. Robinson, Enumeration under two representations of the wreath product, Acta Math., 131 (1973), 123-143.

Cf. A325013 (unoriented), A325014 (chiral), A325015 (achiral), A325016 (exactly k colors).
Other n-dimensional polytopes: A324999 (simplex), A325004 (orthotope).
Rows 1-3 are A000290, A006528, A000543; column 2 is A237748.
Cf. A000048, A001037.

a48[n_] := a48[n] = DivisorSum[NestWhile[#/2&,n,EvenQ], MoebiusMu[#]2^(n/#)&]/(2n); (* A000048 *)
a37[n_] := a37[n] = DivisorSum[n,MoebiusMu[n/#]2^#&]/n; (* A001037 *)
CI0[{n_Integer}] := CI0[{n}] = CI[Transpose[If[EvenQ[n], p2 = IntegerExponent[n, 2]; sub = Divisors[n/2^p2]; {2^(p2+1) sub, a48 /@ (2^p2 sub) }, sub = Divisors[n]; {sub, a37 /@ sub}]]] 2^(n-1);(* even perm. *)
CI1[{n_Integer}] := CI1[{n}] = CI[sub = Divisors[n]; Transpose[If[EvenQ[n], {sub, a37 /@ sub}, {2 sub, a48 /@ sub}]]] 2^(n-1); (* odd perm. *)
compress[x : {{, } ...}] := (s = Sort[x]; For[i = Length[s], i > 1, i -= 1, If[s[[i,1]]==s[[i-1,1]], s[[i-1,2]] += s[[i,2]]; s = Delete[s, i], Null]]; s)
cix[{a_, b_}, {c_, d_}] := {LCM[a, c], (a b c d)/LCM[a, c]};
Unprotect[Times]; Times[CI[a_List], CI[b_List]] :=  (* combine *) CI[compress[Flatten[Outer[cix, a, b, 1], 1]]]; Protect[Times];
CI0[p_List] := CI0[p] = Expand[CI0[Drop[p, -1]] CI0[{Last[p]}] + CI1[Drop[p, -1]] CI1[{Last[p]}]]
CI1[p_List] := CI1[p] = Expand[CI0[Drop[p, -1]] CI1[{Last[p]}] + CI1[Drop[p, -1]] CI0[{Last[p]}]]
pc[p_List] := Module[{ci,mb},mb = DeleteDuplicates[p]; ci = Count[p, #] & /@ mb; n!/(Times @@ (ci!) Times @@ (mb^ci))] (* partition count *)
row[n_Integer] := row[n] = Factor[(Total[(CI0[#] pc[#]) & /@ IntegerPartitions[n]])/(n! 2^(n - 1))] /. CI[l_List] :> j^(Total[l][[2]])
array[n_, k_] := row[n] /. j -> k
Table[array[n, d-n+1], {d, 1, 10}, {n, 1, d}] // Flatten

The algorithm used in the Mathematica program below assigns each permutation of the axes to a partition of n. It then determines the number of permutations for each partition and the cycle index for each partition.
A(n,k) = A325013(n,k) + A325014(n,k) = 2*A325013(n,k) - A325015(n,k) = 2*A325014(n,k) + A325015(n,k).
A(n,k) = Sum_{j=1..2^n} A325016(n,j) * binomial(k,j).

A002076 Number of equivalence classes of base-3 necklaces of length n, where necklaces are considered equivalent under both rotations and permutations of the symbols.

Original entry on oeis.org

1, 1, 2, 3, 6, 9, 26, 53, 146, 369, 1002, 2685, 7434, 20441, 57046, 159451, 448686, 1266081, 3588002, 10195277, 29058526, 83018783, 237740670, 682196949, 1961331314, 5648590737, 16294052602, 47071590147, 136171497650, 394427456121, 1143839943618, 3320824711205

Offset: 0

N. J. A. Sloane

Number of set partitions of an oriented cycle of length n with 3 or fewer subsets. - Robert A. Russell, Nov 05 2018

			E.g., a(2) = 2 as there are two equivalence classes of the 9 strings {00,01,02,10,11,12,20,21,22}: {00,11,22} form one equivalence class and {01,02,10,12,20,21} form the other. To see that (for example) 01 and 02 are equivalent, rotate 01 to 10 and then subtract 1 mod 3 from each element in 10 to get 02.
For a(6)=26, there are 18 achiral patterns (AAAAAA, AAAAAB, AAAABB, AAABAB, AAABBB, AABAAB, AABABB, ABABAB, AAAABC, AAABAC, AAABCB, AABAAC, AABBCC, AABCBC, AABCCB, ABABAC, ABACBC, ABCABC) and 8 chiral patterns in four pairs (AAABBC-AAABCC, AABABC-AABCAC, AABACB-AABCAB, AABACC-AABBAC). - _Robert A. Russell_, Nov 05 2018

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..500
N. J. Fine, Classes of periodic sequences, Illinois J. Math., 2 (1958), 285-302.
E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.
Marko Riedel, Necklaces with swappable colors by Power Group Enumeration
Marko Riedel, Maple code for any necklace size, any number of swappable colors, by Power Group Enumeration.
N. J. A. Sloane, Maple code for this and related sequences
Index entries for sequences related to necklaces

Cf. A000013, A000048, A002075.

Cf. A056353 (unoriented), A320743 (chiral), A182522 (achiral).

Adn[d_, n_] := Module[{ c, t1, t2}, t2 = 0; For[c = 1, c <= d, c++, If[Mod[d, c] == 0 , t2 = t2 + (x^c/c)*(E^(c*z) - 1)]]; t1 = E^t2; t1 = Series[t1, {z, 0, n+1}]; Coefficient[t1, z, n]*n!]; Pn[n_] := Module[{ d, e, t1}, t1 = 0; For[d = 1, d <= n, d++, If[Mod[n, d] == 0, t1 = t1 + EulerPhi[d]*Adn[d, n/d]/n]]; t1/(1 - x)]; Pnq[n_, q_] := Module[{t1}, t1 = Series[Pn[n], {x, 0, q+1}] ; Coefficient[t1, x, q]]; a[n_] := Pnq[n, 3]; Print[1]; Table[Print[an = a[n]]; an, {n, 1, 28}] (* Jean-François Alcover, Oct 04 2013, after N. J. A. Sloane's Maple code *)
(* This Mathematica program uses Gilbert and Riordan's recurrence formula, which they recommend for calculations: *)
Adn[d_, n_] := Adn[d, n] = If[1==n, DivisorSum[d, x^# &],
  Expand[Adn[d, 1] Adn[d, n-1] + D[Adn[d, n-1], x] x]];
Join[{1},Table[SeriesCoefficient[DivisorSum[n, EulerPhi[#] Adn[#, n/#] &] /(n (1 - x)), {x, 0, 3}], {n,40}]]  (* Robert A. Russell, Feb 24 2018 *)
From Robert A. Russell, May 29 2018: (Start)
Join[{1},Table[(1/n) DivisorSum[n, EulerPhi[#] Which[Divisible[#, 6], 3 StirlingS2[n/#+2, 3] - 9 StirlingS2[n/#+1, 3] + 6 StirlingS2[n/#, 3], Divisible[#, 3], 2 StirlingS2[n/#+2, 3] - 7 StirlingS2[n/#+1, 3] + 6 StirlingS2[n/#, 3], Divisible[#, 2], 2 StirlingS2[n/#+2, 3] - 6 StirlingS2[n/#+1, 3] + 4 StirlingS2[n/#, 3], True, StirlingS2[n/#+2, 3] - 4 StirlingS2[n/#+1, 3] + 4 StirlingS2[n/#, 3]] &], {n,40}]] (* or *)
mx = 40; CoefficientList[Series[1 - Sum[(EulerPhi[d] / d) Which[
  Divisible[d, 6], Log[1 - 3x^d], Divisible[d, 3], (Log[1 - 3x^d] +
  Log[1 - x^d]) / 2, Divisible[d, 2], 2 Log[1 - 3x^d] / 3, True, (Log[1 - 3x^d] + 3 Log[1 - x^d]) / 6], {d, 1, mx}], {x, 0, mx}], x]
(End)
(* Adnk(n,d,k) is coefficient of x^k in A(d,n)(x) from Gilbert & Riordan *)
Adnk[d_,n_,k_] := Adnk[d,n,k] = If[n>0 && k>0, Adnk[d,n-1,k]k + DivisorSum[d,Adnk[d,n-1,k-#]&], Boole[n==0 && k==0]]
k=3; Join[{1},Table[Sum[DivisorSum[n,EulerPhi[#] Adnk[#,n/#,j] &],{j,k}]/n,{n,40}]] (* Robert A. Russell, Nov 05 2018 *)

Reference gives formula.
From Robert A. Russell, May 29 2018: (Start)
For n>0, a(n) = (1/n) * Sum_{d|n} phi(d) * ([d==0 mod 6] * (3*S2(n/d+2, 3) - 9*S2(n/d+1, 3) + 6*S2(n/d, 3)) + [d==3 mod 6] * (2*S2(n/d+2, 3) - 7*S2(n/d+1, 3) + 6*S2(n/d, 3)) + [d==2 mod 6 | d==4 mod 6] * (2*S2(n/d+2, 3) - 6*S2(n/d+1, 3) + 4*S2(n/d, 3)) + [d==1 mod 6 | d=5 mod 6] * (S2(n/d+2, 3) - 4*S2(n/d+1, 3) + 4*S2(n/d, 3))), where S2(n,k) is the Stirling subset number, A008277.
G.f.: 1 - Sum_{d>0} (phi(d) / d) * ([d==0 mod 6] * log(1-3x^d) +
  [d==3 mod 6] * (log(1-3x^d) + log(1-x^d)) / 2 +
  [d==2 mod 6 | d==4 mod 6] * 2*log(1-3x^d) / 3 +
  [d==1 mod 6 | d=5 mod 6] * (log(1-3x^d) + 3*log(1-x^d)) / 6).
(End)

Better description and more terms from Mark Weston (mweston(AT)uvic.ca), Oct 06 2001

a(0)=1 prepended by Robert A. Russell, Nov 05 2018

A107424 Triangle read by rows: T(n, k) is the number of primitive (period n) n-bead necklace structures with k different colors. Only includes structures that contain all k colors.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 3, 5, 2, 1, 0, 5, 17, 13, 3, 1, 0, 9, 43, 50, 20, 3, 1, 0, 16, 124, 220, 136, 36, 4, 1, 0, 28, 338, 866, 773, 296, 52, 4, 1, 0, 51, 941, 3435, 4280, 2303, 596, 78, 5, 1, 0, 93, 2591, 13250, 22430, 16317, 5817, 1080, 105, 5, 1, 0, 170, 7234, 51061

Offset: 1

David Wasserman, May 26 2005

This classification is concerned with which beads are the same color, not with the colors themselves, so bbabcd is the same structure as aabacd. Cyclic permutations are also the same structure, e.g. abacda is also the same structure. However, order matters: the reverse of aabacd is equivalent to aabcad, which is also on the list.

			T(6, 4) = 13: {aaabcd, aabacd, aabcad, abacad, aabbcd, aabcbd, aabcdb, aacbbd, aacbdb, ababcd, abacbd, acabdb, abcabd}.
From _Andrew Howroyd_, Apr 09 2017 (Start)
Triangle starts:
1
0  1
0  1   1
0  2   2    1
0  3   5    2    1
0  5  17   13    3    1
0  9  43   50   20    3   1
0 16 124  220  136   36   4  1
0 28 338  866  773  296  52  4 1
0 51 941 3435 4280 2303 596 78 5 1
(End)

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 rows)

Columns 2-6 are A056303, A056304, A056305, A056306, A056307.
Partial row sums include A000048, A002075, A056300, A056301, A056302.
Row sums are A276547.
Cf. A074650, A152175, A276543, A137651, A276544.

A[d_, n_] := A[d, n] = Which[n == 0, 1, n == 1, DivisorSum[d, x^# &], d == 1, Sum[StirlingS2[n, k] x^k, {k, 0, n}], True, Expand[A[d, 1] A[d, n-1] + D[A[d, n-1], x] x]];
B[n_, k_] := Coefficient[DivisorSum[n, EulerPhi[#] A[#, n/#]&]/n/x, x, k];
T[n_, k_] := DivisorSum[n, MoebiusMu[n/#] B[#, k]&];
Table[T[n, k], {n, 1, 12}, {k, 0, n-1}] // Flatten (* Jean-François Alcover, Jun 06 2018, after Andrew Howroyd and Robert A. Russell *)

\\ here R(n) is A152175 as square matrix.
R(n) = {Mat(Col([Vecrev(p/y, n) | p<-Vec(intformal(sum(m=1, n, eulerphi(m) * subst(serlaplace(-1 + exp(sumdiv(m, d, y^d*(exp(d*x + O(x*x^(n\m)))-1)/d))), x, x^m))/x))]))}
T(n) = {my(M=R(n)); matrix(n, n, i, k, sumdiv(i, d, moebius(i/d)*M[d,k]))}
{ my(A=T(10)); for(n=1, #A, print(A[n, 1..n])) } \\ Andrew Howroyd, Jan 09 2020

T(n, k) = Sum_{d|n} mu(n/d) * A152175(d, k). - Andrew Howroyd, Apr 09 2017

A325014 Array read by descending antidiagonals: A(n,k) is the number of chiral pairs of colorings of the facets of a regular n-dimensional orthoplex using up to k colors.

Original entry on oeis.org

0, 1, 0, 3, 0, 0, 6, 3, 1, 0, 10, 15, 66, 94, 0, 15, 45, 920, 97974, 1047816, 0, 21, 105, 6350, 10700090, 481141220994, 400140831558512, 0, 28, 210, 29505, 390081800, 4802390808840576, 74515656021475803734579625, 527471421741473576372948457251328, 0

Offset: 1

Robert A. Russell, May 27 2019

Also called cross polytope and hyperoctahedron. For n=1, the figure is a line segment with two vertices. For n=2 the figure is a square with four edges. For n=3 the figure is an octahedron with eight triangular faces. For n=4, the figure is a 16-cell with sixteen tetrahedral facets. The Schläfli symbol, {3,...,3,4}, of the regular n-dimensional orthoplex (n>1) consists of n-2 threes followed by a four. Each of its 2^n facets is an (n-1)-dimensional simplex. The chiral colorings of its facets come in pairs, each the reflection of the other.

Also the number of chiral pairs of colorings of the vertices of a regular n-dimensional orthotope (cube) using up to k colors.

			Array begins with A(1,1):
0  1     3        6        10         15          21           28 ...
0  0     3       15        45        105         210          378 ...
0  1    66      920      6350      29505      106036       317856 ...
0 94 97974 10700090 390081800 7280687610 86121007714 730895668104 ...
For A(2,3)=3, each square has one of the three colors on two adjacent edges.

Robert A. Russell, Table of n, a(n) for n = 1..78
E. M. Palmer and R. W. Robinson, Enumeration under two representations of the wreath product, Acta Math., 131 (1973), 123-143.
Wikipedia, Cross-polytope

Cf. A325012 (oriented), A325013 (unoriented), A325015 (achiral), A325018 (exactly k colors).
Other n-dimensional polytopes: A007318(k,n+1) (simplex), A325006 (orthotope).
Rows 1-2 are A161680, A050534.
Cf. A000048, A001037.

a48[n_] := a48[n] = DivisorSum[NestWhile[#/2&, n, EvenQ], MoebiusMu[#]2^(n/#)&]/(2n); (* A000048 *)
a37[n_] := a37[n] = DivisorSum[n, MoebiusMu[n/#]2^#&]/n; (* A001037 *)
CI0[{n_Integer}] := CI0[{n}] = CI[Transpose[If[EvenQ[n], p2 = IntegerExponent[n, 2]; sub = Divisors[n/2^p2]; {2^(p2+1) sub, a48 /@ (2^p2 sub) }, sub = Divisors[n]; {sub, a37 /@ sub}]]] 2^(n-1); (* even perm. *)
CI1[{n_Integer}] := CI1[{n}] = CI[sub = Divisors[n]; Transpose[If[EvenQ[n], {sub, a37 /@ sub}, {2 sub, a48 /@ sub}]]] 2^(n-1); (* odd perm. *)
compress[x : {{, } ...}] := (s = Sort[x]; For[i = Length[s], i > 1, i -= 1, If[s[[i, 1]]==s[[i-1, 1]], s[[i-1, 2]] += s[[i, 2]]; s = Delete[s, i], Null]]; s)
cix[{a_, b_}, {c_, d_}] := {LCM[a, c], (a b c d)/LCM[a, c]};
Unprotect[Times]; Times[CI[a_List], CI[b_List]] :=  (* combine *) CI[compress[Flatten[Outer[cix, a, b, 1], 1]]]; Protect[Times];
CI0[p_List] := CI0[p] = Expand[CI0[Drop[p, -1]] CI0[{Last[p]}] + CI1[Drop[p, -1]] CI1[{Last[p]}]]
CI1[p_List] := CI1[p] = Expand[CI0[Drop[p, -1]] CI1[{Last[p]}] + CI1[Drop[p, -1]] CI0[{Last[p]}]]
pc[p_List] := Module[{ci,mb},mb = DeleteDuplicates[p]; ci = Count[p, #] & /@ mb; n!/(Times @@ (ci!) Times @@ (mb^ci))] (* partition count *)
row[n_Integer] := row[n] = Factor[(Total[((CI0[#] - CI1[#]) pc[#]) & /@ IntegerPartitions[n]])/(n! 2^n)] /. CI[l_List] :> j^(Total[l][[2]])
array[n_, k_] := row[n] /. j -> k
Table[array[n, d-n+1], {d, 1, 10}, {n, 1, d}] // Flatten

The algorithm used in the Mathematica program below assigns each permutation of the axes to a partition of n. It then determines the number of permutations for each partition and the cycle index for each partition.
A(k,n) = A325012(n,k) - A325013(n,k) = (A325012(n,k) - A325015(n,k)) / 2 = A325013(n,k) - A325015(n,k).
A(n,k) = Sum_{j=2..2^n} A325018(n,j) * binomial(k,j).

cp's OEIS Frontend

A000016 a(n) is the number of distinct (infinite) output sequences from binary n-stage shift register which feeds back the complement of the last stage.

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A000013 Definition (1): Number of n-bead binary necklaces with beads of 2 colors where the colors may be swapped but turning over is not allowed.

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A001372 Number of unlabeled mappings (or mapping patterns) from n points to themselves; number of unlabeled endofunctions.

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A051841 Number of binary Lyndon words with an even number of 1's.

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A054660 Number of monic irreducible polynomials over GF(4) of degree n with fixed nonzero trace.

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A000046 Number of primitive n-bead necklaces (turning over is allowed) where complements are equivalent.

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