A052823 -id:A052823 - cp's OEIS frontend

A000031 Number of n-bead necklaces with 2 colors when turning over is not allowed; also number of output sequences from a simple n-stage cycling shift register; also number of binary irreducible polynomials whose degree divides n.

1, 2, 3, 4, 6, 8, 14, 20, 36, 60, 108, 188, 352, 632, 1182, 2192, 4116, 7712, 14602, 27596, 52488, 99880, 190746, 364724, 699252, 1342184, 2581428, 4971068, 9587580, 18512792, 35792568, 69273668, 134219796, 260301176, 505294128, 981706832

Offset: 0

N. J. A. Sloane

Also a(n)-1 is the number of 1's in the truth table for the lexicographically least de Bruijn cycle (Fredricksen).
In music, a(n) is the number of distinct classes of scales and chords in an n-note equal-tempered tuning system. - Paul Cantrell, Dec 28 2011
Also, minimum cardinality of an unavoidable set of length-n binary words (Champarnaud, Hansel, Perrin). - Jeffrey Shallit, Jan 10 2019
(1/n) * Dirichlet convolution of phi(n) and 2^n, n>0. - Richard L. Ollerton, May 06 2021
From Jianing Song, Nov 13 2021: (Start)
a(n) is even for n != 0, 2. Proof: write n = 2^e * s with odd s, then a(n) * s = Sum_{d|s} Sum_{k=0..e} phi((2^e*s)/(2^k*d)) * 2^(2^k*d-e) = Sum_{d|s} Sum_{k=0..e-1} phi(s/d) * 2^(2^k*d-k-1) + Sum_{d|s} phi(s/d) * 2^(2^e*d-e) == Sum_{k=0..e-1} 2^(2^k*s-k-1) + 2^(2^e*s-e) == Sum_{k=0..min{e-1,1}} 2^(2^k*s-k-1) (mod 2). a(n) is odd if and only if s = 1 and e-1 = 0, or n = 2.
a(n) == 2 (mod 4) if and only if n = 1, 4 or n = 2*p^e with prime p == 3 (mod 4).
a(n) == 4 (mod 8) if and only if n = 2^e, 3*2^e for e >= 3, or n = p^e, 4*p^e != 12 with prime p == 3 (mod 4), or n = 2s where s is an odd number such that phi(s) == 4 (mod 8). (End)

			For n=3 and n=4 the necklaces are {000,001,011,111} and {0000,0001,0011,0101,0111,1111}.
The analogous shift register sequences are {000..., 001001..., 011011..., 111...} and {000..., 00010001..., 00110011..., 0101..., 01110111..., 111...}.

S. W. Golomb, Shift-Register Sequences, Holden-Day, San Francisco, 1967, pp. 120, 172.
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
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).
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 7.112(a).

Seiichi Manyama, Table of n, a(n) for n = 0..3333 (first 201 terms from T. D. Noe)
Nicolás Álvarez, Verónica Becher, Martín Mereb, Ivo Pajor, and Carlos Miguel Soto, On extremal factors of de Bruijn-like graphs, arXiv:2308.16257 [math.CO], 2023. See references.
Joerg Arndt, Matters Computational (The Fxtbook), p. 151, pp. 379-383.
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
J.-M. Champarnaud, G. Hansel, and D. Perrin, Unavoidable sets of constant length, Internat. J. Alg. Comput. 14 (2004), 241-251.
Vladimir Dotsenko and Irvin Roy Hentzel, On the conjecture of Kashuba and Mathieu about free Jordan algebras, arXiv:2507.00437 [math.RA], 2025. See p. 14.
James East and Ron Niles, Integer polygons of given perimeter, arXiv:1710.11245 [math.CO], 2017.
S. N. Ethier and J. Lee, Parrondo games with spatial dependence, arXiv preprint arXiv:1202.2609 [math.PR], 2012.
S. N. Ethier, Counting toroidal binary arrays, arXiv preprint arXiv:1301.2352 [math.CO], 2013 and J. Int. Seq. 16 (2013) #13.4.7.
N. J. Fine, Classes of periodic sequences, Illinois J. Math., 2 (1958), 285-302.
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see pages 18, 64.
H. Fredricksen, The lexicographically least de Bruijn cycle, J. Combin. Theory, 9 (1970) 1-5.
Harold Fredricksen, An algorithm for generating necklaces of beads in two colors, Discrete Mathematics, Volume 61, Issues 2-3, September 1986, Pages 181-188.
E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 2
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 130
Juhani Karhumäki, S. Puzynina, M. Rao, and M. A. Whiteland, On cardinalities of k-abelian equivalence classes, arXiv preprint arXiv:1605.03319 [math.CO], 2016.
Abraham Lempel, On extremal factors of the de Bruijn graph, J. Combinatorial Theory Ser. B 11 1971 17--27. MR0276126 (43 #1874).
Karyn McLellan, Periodic coefficients and random Fibonacci sequences, Electronic Journal of Combinatorics, 20(4), 2013, #P32.
Johannes Mykkeltveit, A proof of Golomb's conjecture for the de Bruijn graph, J. Combinatorial Theory Ser. B 13 (1972), 40-45. MR0323629 (48 #1985).
Matthew Parker, The first 25K terms (7-Zip compressed file) [a large file]
J. Riordan, Letter to N. J. A. Sloane, Jul. 1978
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]
Ville Salo, Universal gates with wires in a row, arXiv:1809.08050 [math.GR], 2018.
J. A. Siehler, The Finite Lamplighter Groups: A Guided Tour, College Mathematics Journal, Vol. 43, No. 3 (May 2012), pp. 203-211.
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.
David Thomson, Musical Polygons, Mathematics Today, Vol. 57, No. 2 (April 2021), pp. 50-51.
R. C. Titsworth, Equivalence classes of periodic sequences, Illinois J. Math., 8 (1964), 266-270.
A. M. Uludag, A. Zeytin and M. Durmus, Binary Quadratic Forms as Dessins, 2012.
Eric Weisstein's World of Mathematics, Necklace
Wolfram Research, Number of necklaces
Index entries for "core" sequences
Index entries for sequences related to necklaces

Column 2 of A075195.
Cf. A001037 (primitive solutions to same problem), A014580, A000016, A000013, A000029 (if turning over is allowed), A000011, A001371, A058766.
Rows sums of triangle in A047996.
Dividing by 2 gives A053634.
A008965(n) = a(n) - 1 allowing different offsets.
Cf. A008965, A053635, A052823, A100447 (bisection).
Cf. A000010.

a000031 0 = 1
a000031 n = (`div` n) $ sum $
   zipWith (*) (map a000010 divs) (map a000079 $ reverse divs)
   where divs = a027750_row n
-- Reinhard Zumkeller, Mar 21 2013

with(numtheory); A000031 := proc(n) local d,s; if n = 0 then RETURN(1); else s := 0; for d in divisors(n) do s := s+phi(d)*2^(n/d); od; RETURN(s/n); fi; end; [ seq(A000031(n), n=0..50) ];

a[n_] := Sum[If[Mod[n, d] == 0, EulerPhi[d] 2^(n/d), 0], {d, 1, n}]/n
a[n_] := Fold[#1 + 2^(n/#2) EulerPhi[#2] &, 0, Divisors[n]]/n (* Ben Branman, Jan 08 2011 *)
Table[Expand[CycleIndex[CyclicGroup[n], t] /. Table[t[i]-> 2, {i, 1, n}]], {n,0, 30}] (* Geoffrey Critzer, Mar 06 2011*)
a[0] = 1; a[n_] := DivisorSum[n, EulerPhi[#]*2^(n/#)&]/n; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 03 2016 *)
mx=40; CoefficientList[Series[1-Sum[EulerPhi[i] Log[1-2*x^i]/i,{i,1,mx}],{x,0,mx}],x] (*Herbert Kociemba, Oct 29 2016 *)

{A000031(n)=if(n==0,1,sumdiv(n,d,eulerphi(d)*2^(n/d))/n)} \\ Randall L Rathbun, Jan 11 2002

from sympy import totient, divisors
def A000031(n): return sum(totient(d)*(1<Chai Wah Wu, Nov 16 2022

a(n) = (1/n)*Sum_{ d divides n } phi(d)*2^(n/d) = A053635(n)/n, where phi is A000010.
Warning: easily confused with A001037, which has a similar formula.
G.f.: 1 - Sum_{n>=1} phi(n)*log(1 - 2*x^n)/n. - Herbert Kociemba, Oct 29 2016
a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} 2^gcd(n,k). - Ilya Gutkovskiy, Apr 16 2021
a(0) = 1; a(n) = (1/n)*Sum_{k=1..n} 2^(n/gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). - Richard L. Ollerton, May 06 2021
Dirichlet g.f.: f(s+1) * (zeta(s)/zeta(s+1)), where f(s) = Sum_{n>=1} 2^n/n^s. - Jianing Song, Nov 13 2021

There is an error in Fig. M3860 in the 1995 Encyclopedia of Integer Sequences: in the third line, the formula for A000031 = M0564 should be (1/n) sum phi(d) 2^(n/d).

A087854 Triangle read by rows: T(n,k) is the number of n-bead necklaces with exactly k different colored beads.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 4, 9, 6, 1, 6, 30, 48, 24, 1, 12, 91, 260, 300, 120, 1, 18, 258, 1200, 2400, 2160, 720, 1, 34, 729, 5106, 15750, 23940, 17640, 5040, 1, 58, 2018, 20720, 92680, 211680, 258720, 161280, 40320, 1, 106, 5613, 81876, 510312, 1643544, 2963520, 3024000, 1632960, 362880

Offset: 1

Philippe Deléham

Equivalently, T(n,k) is the number of sequences (words) of length n on an alphabet of k letters where each letter of the alphabet occurs at least once in the sequence. Two sequences are considered equivalent if one can be obtained from the other by a cyclic shift of the letters. Cf. A054631 where the surjective restriction is removed. - Geoffrey Critzer, Jun 18 2013

Robert A. Russell's g.f. for column k >= 1 (in the Formula section below) can be proved by integrating both sides of the formula Sum_{n>=1} S2(n, k)*x^(n-1) = x^(k-1)/((1 - x)* (1 - 2*x) * (1 - 3*x) * ... * (1 - k*x)) w.r.t. x. A variation of this identity (valid for |x| < 1/k) can be found in the Formula section of A008277. - Petros Hadjicostas, Aug 20 2019

			The triangle begins with T(1,1):
  1;
  1,   1;
  1,   2,    2;
  1,   4,    9,     6;
  1,   6,   30,    48,     24;
  1,  12,   91,   260,    300,     120;
  1,  18,  258,  1200,   2400,    2160,     720;
  1,  34,  729,  5106,  15750,   23940,   17640,    5040;
  1,  58, 2018, 20720,  92680,  211680,  258720,  161280,   40320;
  1, 106, 5613, 81876, 510312, 1643544, 2963520, 3024000, 1632960, 362880;
  ...
For T(4,2) = 4, the necklaces are AAAB, AABB, ABAB, and ABBB.
For T(4,4) = 6, the necklaces are ABCD, ABDC, ACBD, ACDB, ADBC, and ADCB.

Alois P. Heinz, Rows n = 1..141, flattened

Columns 1-6: A057427, A052823, A056283, A056284, A056285, A056286.
Diagonals: A000142 and A074143.
Row sums: A019536.
Cf. A000010 (Euler totient phi function), A008277 (Stirling2 numbers), A075195 (table of Jablonski).
Cf. A254040, A273891.

with(numtheory):
T:= (n, k)-> (k!/n) *add(phi(d) *Stirling2(n/d, k), d=divisors(n)):
seq(seq(T(n,k), k=1..n), n=1..12);  # Alois P. Heinz, Jun 19 2013

Table[Table[Sum[EulerPhi[d]*StirlingS2[n/d,k]k!,{d,Divisors[n]}]/n,{k,1,n}],{n,1,10}]//Grid (* Geoffrey Critzer, Jun 18 2013 *)

T(n, k) = (k!/n) * sumdiv(n, d, eulerphi(d) * stirling(n/d, k, 2)); \\ Joerg Arndt, Sep 25 2020

T(n,k) = Sum_{i=0..k-1} (-1)^i * C(k,i) * A075195(n,k-i); A075195 = Jablonski's table.
T(n,k) = (k!/n) * Sum_{d|n} phi(d) * S2(n/d, k), where S2(n,k) = Stirling numbers of 2nd kind A008277.
G.f. for column k: -Sum_{d>0} (phi(d)/d) * Sum_{j = 1..k} (-1)^(k-j) * C(k,j) * log(1 - j * x^d). - Robert A. Russell, Sep 26 2018
T(n,k) = Sum_{d|n} A254040(d, k) for n, k >= 1. - Petros Hadjicostas, Aug 19 2019

Formula section edited by Petros Hadjicostas, Aug 20 2019

A059076 Number of pairs of orientable necklaces with n beads and two colors; i.e., turning the necklace over does not leave it unchanged.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 2, 6, 14, 30, 62, 128, 252, 495, 968, 1866, 3600, 6917, 13286, 25476, 48916, 93837, 180314, 346554, 666996, 1284570, 2477342, 4781502, 9240012, 17871708, 34604066, 67060746, 130085052, 252548760, 490722344

Offset: 0

Henry Bottomley, Dec 22 2000

nonn

Number of chiral bracelets with n beads and two colors.

			For n=6, the only chiral pair is AABABB-AABBAB.  For n=7, the two chiral pairs are AAABABB-AAABBAB and AABABBB-AABBBAB. - _Robert A. Russell_, Sep 24 2018

G. C. Greubel, Table of n, a(n) for n = 0..1000
Daniel Gabric and Joe Sawada, Efficient Construction of Long Orientable Sequences, arXiv:2401.14341 [cs.DS], 2024.
Petros Hadjicostas, Formulas for chiral bracelets, 2019; see Section 5.
John P. McSorley and Alan H. Schoen, Rhombic tilings of (n,k)-ovals, (n, k, lambda)-cyclic difference sets, and related topics, Discrete Math., 313 (2013), 129-154. - From _N. J. A. Sloane_, Nov 26 2012

Column 2 of A293496.
Cf. A059053.
Column 2 of A305541.
Equals (A000031 - A164090) / 2.
a(n) = (A052823(n) - A027383(n-2)) / 2.

nn=35;Table[CoefficientList[Series[CycleIndex[CyclicGroup[n],s]-CycleIndex[DihedralGroup[n],s]/.Table[s[i]->2,{i,1,n}],{x,0,nn}],x],{n,1,nn}]//Flatten  (* Geoffrey Critzer, Mar 26 2013 *)
mx=40; CoefficientList[Series[(1-Sum[ EulerPhi[n]*Log[1-2*x^n]/n, {n, mx}]-(1+x)^2/(1-2*x^2))/2, {x, 0, mx}], x] (* Herbert Kociemba, Nov 02 2016 *)
terms = 36; a29[0] = 1; a29[n_] := (1/4)*(Mod[n, 2] + 3)*2^Quotient[n, 2] + DivisorSum[n, EulerPhi[#]*2^(n/#) & ]/(2*n); Array[a29, 36, 0] - LinearRecurrence[{0, 2}, {1, 2, 3}, 36] (* Jean-François Alcover, Nov 05 2017 *)
k = 2; Prepend[Table[DivisorSum[n, EulerPhi[#] k^(n/#) &]/(2n)(k^Floor[(n+1)/2] + k^Ceiling[(n+1)/2])/4, {n, 1, 30}], 0] (* Robert A. Russell, Sep 24 2018 *)

a(n) = A000031(n) - A000029(n) = A000029(n) - A029744(n) = (A000031(n) - A029744(n))/2 = A008965(n) - A091696(n)
G.f.: (1 - Sum_{n>=1} phi(n)*log(1 - 2*x^n)/n - (1 + x)^2/(1 - 2*x^2))/2. - Herbert Kociemba, Nov 02 2016
For n > 0, a(n) = -(k^floor((n + 1)/2) + k^ceiling((n + 1)/2))/4 + (1/(2*n))* Sum_{d|n} phi(d)*k^(n/d), where k = 2 is the maximum number of colors. - Robert A. Russell, Sep 24 2018

A056295 Number of n-bead necklace structures using exactly two different colored beads.

Original entry on oeis.org

0, 1, 1, 3, 3, 7, 9, 19, 29, 55, 93, 179, 315, 595, 1095, 2067, 3855, 7315, 13797, 26271, 49939, 95419, 182361, 349715, 671091, 1290871, 2485533, 4794087, 9256395, 17896831, 34636833, 67110931, 130150587, 252648991, 490853415, 954444607, 1857283155, 3616828363

Offset: 1

Marks R. Nester

Turning over the necklace is not allowed. Colors may be permuted without changing the necklace structure.

			For a(7) = 9, the color patterns are AAAAAAB, AAAAABB, AAAABAB, AAAABBB, AAABAAB, AABAABB, AABABAB, AAABABB, and AAABBAB. The first seven are achiral; the last two are a chiral pair. - _Robert A. Russell_, Mar 08 2018

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2.]

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
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.

Column 2 of A152175.

Cf. A000013, A052823.

Maple
```
See A000013.
```

Table[DivisorSum[n, EulerPhi[#] If[OddQ[#], StirlingS2[n/#, 2], StirlingS2[n/#+1, 2]]&]/n, {n,1,30}] (* Robert A. Russell, Feb 20 2018 *)

a(n) = A000013(n) - 1.
From Robert A. Russell, Mar 08 2018: (Start)
G.f.: Sum_{ d>0 } phi(d)*(2*log(1-x^d) - (1+[d == 0 mod 2])*log(1-2*x^d)) / (2*d);
a(n) = (1/n)*Sum_{d|n} phi(d) * S2(n/d + [d == 0 mod 2], 2), where S2(n, k) is the Stirling subset number, A008277. (End)

A294684 Triangle read by rows: T(n,k) is the number of non-isomorphic colorings of a toroidal n X k grid using exactly two colors under translational symmetry, 1 <= k <= n.

Original entry on oeis.org

0, 1, 5, 2, 12, 62, 4, 38, 350, 4154, 6, 106, 2190, 52486, 1342206, 12, 360, 14622, 699598, 35792566, 1908897150, 18, 1180, 99878, 9587578, 981706830, 104715443850, 11488774559742, 34, 4148, 699250, 134223974, 27487816990, 5864063066498, 1286742755471398, 288230376353050814

Offset: 1

Marko Riedel, Nov 06 2017

Colors are not being permuted, i.e., Power Group Enumeration does not apply here.

			Triangle begins:
   0;
   1,    5;
   2,   12,    62;
   4,   38,   350,    4154;
   6,  106,  2190,   52486,   1342206;
  12,  360, 14622,  699598,  35792566,   1908897150;
  18, 1180, 99878, 9587578, 981706830, 104715443850, 11488774559742;
  ...
For the 2 X 2 and two colors we find
  +---+  +---+  +---+  +---+  +---+
  |X| |  | |X|  |X| |  |X|X|  |X| |
  +-+-+  +-+-+  +-+-+  +-+-+  +-+-+
  | | |  |X|X|  | |X|  | | |  |X| |
  +-+-+  +-+-+  +-+-+  +-+-+  +-+-+

F. Harary and E. Palmer, Graphical Enumeration, Academic Press, 1973.

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 rows)
Marko Riedel et al., Burnside lemma and translational symmetries of the torus.
Marko Riedel, Maple code for sequences A294684, A294685, A294686, A294687.

Main diagonal is A376822.

Cf. A184271, A294685, A294686, A294687, A294791, A294792, A294793, A294794. T(n,1) is A052823.

With[{Q = 2}, Table[(Q!/n/k) Sum[Sum[EulerPhi[d] EulerPhi[f] StirlingS2[GCD[d, f] (n/d) (k/f), Q], {f, Divisors@ k}], {d, Divisors@ n}], {n, 8}, {k, n}]] // Flatten (* Michael De Vlieger, Nov 08 2017 *)

T(n,m) = {2*sumdiv(n, d, sumdiv(m, e, eulerphi(d) * eulerphi(e) * stirling(n*m/lcm(d,e), 2, 2) ))/(n*m)} \\ Andrew Howroyd, Oct 05 2024

T(n,k) = (Q!/(n*k))*(Sum_{d|n} Sum_{f|k} phi(d) phi(f) S(gcd(d,f)*(n/d)*(k/f), Q)) with Q=2 and S(n,k) Stirling numbers of the second kind.

T(n,k) = A184271(n,k) - 2. - Andrew Howroyd, Oct 05 2024

A056283 Number of n-bead necklaces with exactly three different colored beads.

Original entry on oeis.org

0, 0, 2, 9, 30, 91, 258, 729, 2018, 5613, 15546, 43315, 120750, 338259, 950062, 2678499, 7573350, 21480739, 61088874, 174184755, 497812638, 1425847623, 4092087522, 11765822365, 33887517870, 97756387365, 282414624746, 816999710223, 2366509198350, 6862930841141

Offset: 1

Marks R. Nester

nonn

Turning over the necklace is not allowed.

			For n=3, the two necklaces are ABC and ACB.

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A000031, A001867, A052823.

Column k=3 of A087854.

k=3; Table[k!DivisorSum[n,EulerPhi[#]StirlingS2[n/#,k]&]/n,{n,1,30}] (* Robert A. Russell, Sep 26 2018 *)

a(n) = A001867(n) - 3*A000031(n) + 3.
From Robert A. Russell, Sep 26 2018: (Start)
a(n) = (k!/n) Sum_{d|n} phi(d) S2(n/d,k), where k=3 is the number of colors and S2 is the Stirling subset number A008277.
G.f.: -Sum_{d>0} (phi(d)/d) * Sum_{j} (-1)^(k-j) * C(k,j) * log(1-j x^d), where k=3 is the number of colors. (End)

A056342 Number of bracelets of length n using exactly two different colored beads.

Original entry on oeis.org

0, 1, 2, 4, 6, 11, 16, 28, 44, 76, 124, 222, 378, 685, 1222, 2248, 4110, 7683, 14308, 27010, 50962, 96907, 184408, 352696, 675186, 1296856, 2493724, 4806076, 9272778, 17920858, 34669600, 67159048, 130216122, 252745366, 490984486, 954637556, 1857545298, 3617214679, 7048675958, 13744694926, 26818405350

Offset: 1

Marks R. Nester

nonn

Turning over will not create a new bracelet.

			For a(6)=11, the arrangements are AAAAAB, AAAABB, AAABAB, AAABBB, AABAAB, AABBBB, ABABAB, ABABBB, ABBABB, ABBBBB, and AABABB, the last being chiral. Its reverse is AABBAB. - _Robert A. Russell_, Sep 26 2018

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

G. C. Greubel, Table of n, a(n) for n = 1..3000

Column 2 of A273891.
Equals A052823 - A059076.
Cf. A008277, A027383.

a[n_] := (1/4)*(Mod[n, 2] + 3)*2^Quotient[n, 2] + DivisorSum[n, EulerPhi[#]*2^(n/#)&]/(2*n) - 2; Array[a, 41] (* Jean-François Alcover, Nov 05 2017 *)
k=2; Table[k! DivisorSum[n, EulerPhi[#] StirlingS2[n/#,k]&]/(2n) + k!(StirlingS2[Floor[(n+1)/2], k] + StirlingS2[Ceiling[(n+1)/2], k])/4, {n,1,30}] (* Robert A. Russell, Sep 26 2018 *)

a(n) = my(k=2); (k!/4)*(stirling(floor((n+1)/2),k,2) + stirling(ceil((n+1)/2),k,2)) + (k!/(2*n))*sumdiv(n,d,eulerphi(d)*stirling(n/d,k,2)); \\ Michel Marcus, Sep 28 2018

a(n) = A000029(n) - 2.
From Robert A. Russell, Sep 26 2018: (Start)
a(n) = (A052823(n) + A027383(n-2)) / 2 = A059076(n) + A027383(n-2).
a(n) = (k!/4) * (S2(floor((n+1)/2),k) + S2(ceiling((n+1)/2),k)) + (k!/2n) * Sum_{d|n} phi(d) * S2(n/d,k), where k=2 is the number of colors and S2 is the Stirling subset number A008277.
G.f.: (k!/4) * x^(2k-2) * (1+x)^2 / Product_{i=1..k} (1-i x^2) - Sum_{d>0} (phi(d)/2d) * Sum_{j} (-1)^(k-j) * C(k,j) * log(1-j x^d), where k=2 is the number of colors. (End)

More terms from Joerg Arndt, Jun 10 2016

cp's OEIS Frontend

A000031 Number of n-bead necklaces with 2 colors when turning over is not allowed; also number of output sequences from a simple n-stage cycling shift register; also number of binary irreducible polynomials whose degree divides n.

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A087854 Triangle read by rows: T(n,k) is the number of n-bead necklaces with exactly k different colored beads.

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A059076 Number of pairs of orientable necklaces with n beads and two colors; i.e., turning the necklace over does not leave it unchanged.

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A056295 Number of n-bead necklace structures using exactly two different colored beads.

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A294684 Triangle read by rows: T(n,k) is the number of non-isomorphic colorings of a toroidal n X k grid using exactly two colors under translational symmetry, 1 <= k <= n.

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A056283 Number of n-bead necklaces with exactly three different colored beads.

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