A001868 -id:A001868 - cp's OEIS frontend

A075195 Jablonski table T(n,k) read by antidiagonals: T(n,k) = number of necklaces with n beads of k colors.

1, 2, 1, 3, 3, 1, 4, 6, 4, 1, 5, 10, 11, 6, 1, 6, 15, 24, 24, 8, 1, 7, 21, 45, 70, 51, 14, 1, 8, 28, 76, 165, 208, 130, 20, 1, 9, 36, 119, 336, 629, 700, 315, 36, 1, 10, 45, 176, 616, 1560, 2635, 2344, 834, 60, 1, 11, 55, 249, 1044, 3367, 7826, 11165, 8230, 2195, 108, 1

Offset: 1

Christian G. Bower, Sep 07 2002

From Richard L. Ollerton, May 07 2021: (Start)
Here, as in A000031, turning over is not allowed.
(1/n) * Dirichlet convolution of phi(n) and k^n. (End)

			The array T(n,k) for n >= 1, k >= 1 begins:
  1,  2,   3,    4,     5,     6,      7, ...
  1,  3,   6,   10,    15,    21,     28, ...
  1,  4,  11,   24,    45,    76,    119, ...
  1,  6,  24,   70,   165,   336,    616, ...
  1,  8,  51,  208,   629,  1560,   3367, ...
  1, 14, 130,  700,  2635,  7826,  19684, ...
  1, 20, 315, 2344, 11165, 39996, 117655, ...
From _Indranil Ghosh_, Mar 25 2017: (Start)
Triangle formed when the array is read by antidiagonals:
   1;
   2,  1;
   3,  3,   1;
   4,  6,   4,   1;
   5, 10,  11,   6,    1;
   6, 15,  24,  24,    8,    1;
   7, 21,  45,  70,   51,   14,    1;
   8, 28,  76, 165,  208,  130,   20,   1;
   9, 36, 119, 336,  629,  700,  315,  36,  1;
  10, 45, 176, 616, 1560, 2635, 2344, 834, 60, 1;
  ... (End)

F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Cambridge, 1998, p. 86 (2.2.23).
Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 496.
Louis Comtet, Analyse combinatoire, Tome 2, p. 104 #17, P.U.F., 1970.

Andrew Howroyd, Table of n, a(n) for n = 1..1275
E. Jablonski, Théorie des permutations et des arrangements complets, Journal de Liouville, 8 (1892), pp. 331-49.
E. Jablonski, Sur l'analyse combinatoire circulaire, Comptes-rendus de l'Académie des Sciences, April 11 1892, Paris.
E. Lucas, Sur les permutations circulaires avec répétition, Théorie des Nombres, Annexe VII, Gauthier-Villars, Paris, 1891. Mentions Moreau.
P. A. MacMahon, Application of a theory of permutations in circular procession to the theory of numbers, Proc. Lond. Math. Soc., 23 (1892), pp. 305-313. Mentions Jablonski, Lucas and Moreau.
Index entries for sequences related to necklaces

Columns 1-10: A000012, A000031, A001867, A001868, A001869, A054625-A054629.
Rows 1-10: A000027, A000217, A006527, A006528, A054620, A006565, A054621-A054624.
Main Diagonal: A056665. A054630 and A054631 are the upper and lower triangles.
Cf. A000010, A185651, A246935.

t[n_, k_] := (1/n)*Sum[EulerPhi[d]*k^(n/d), {d, Divisors[n]}]; Table[t[n-k+1, k], {n, 1, 11}, {k, n, 1, -1}] // Flatten (* Jean-François Alcover, Jan 20 2014, after Philippe Deléham *)

T(n, k) = (1/n) * sumdiv(n, d, eulerphi(d)*k^(n/d));
for(n=1, 15, for(k=1, n, print1(T(k, n - k + 1),", ");); print();) \\ Indranil Ghosh, Mar 25 2017

from sympy.ntheory import totient, divisors
def T(n,k): return sum(totient(d)*k**(n//d) for d in divisors(n))//n
for n in range(1, 16):
    print([T(k, n - k + 1) for k in range(1, n + 1)]) # Indranil Ghosh, Mar 25 2017

T(n,k) = (1/n)*Sum_{d | n} phi(d)*k^(n/d), where phi = Euler totient function A000010. - Philippe Deléham, Oct 08 2003
From Petros Hadjicostas, Feb 08 2021: (Start)
O.g.f. for column k >= 1: Sum_{n>=1} T(n,k)*x^n = -Sum_{j >= 1} (phi(j)/j) * log(1 - k*x^j).
Linear recurrence for row n >= 1: T(n,k) = Sum_{j=0..n} -binomial(j-n-1,j+1) * T(n,k-1-j) for k >= n + 2. (This recurrence is essentially due to Robert A. Russell, who contributed it in A321791.) (End)
From Richard L. Ollerton, May 07 2021: (Start)
T(n,k) = (1/n)*Sum_{i=1..n} k^gcd(n,i).
T(n,k) = (1/n)*Sum_{i=1..n} k^(n/gcd(n,i))*phi(gcd(n,i))/phi(n/gcd(n,i)).
T(n,k) = (1/n)*A185651(n,k) for n >= 1, k >= 1. (End)
Product_{n>=1} 1/(1 - x^n)^T(n,k) = Product_{n>=1} 1/(1 - k*x^n). - Seiichi Manyama, Apr 12 2025

Additional references from Philippe Deléham, Oct 08 2003

A067855 Square of the Euclidean length of the vector of Littlewood-Richardson coefficients of Sum_{lambda |- n} s_lambda^2, where s_lambda are the symmetric Schur functions and the sum runs over all partitions lambda of n.

Original entry on oeis.org

1, 2, 8, 26, 94, 326, 1196, 4358, 16248, 60854, 230184, 874878, 3343614, 12825418, 49368388, 190554410, 737328366, 2858974502, 11106267880, 43215101102, 168398785002, 657070401106, 2566847255572, 10038191414610, 39295007540748

Offset: 0

Richard Stanley, Feb 15 2002

Original name: "Squared length of sum of s_lambda^2, where s_lambda is a Schur function and lambda ranges over all partitions of n."
This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = 1/2, g(n) = 4. - Seiichi Manyama, Apr 22 2018
The symbol "|-" means "is a partition of", cf. MathWorld link and the Geloun & Ramgoolam paper. The Littlewood-Richardson coefficients allow a product of two Schur functions to be expressed as a linear combination of Schur functions of the corresponding degree. (The Schur functions symmetric in all n variables correspond to Schur polynomials of partitions extended with 0's to length n.) - M. F. Hasler, Jan 19 2020
See A070933 for similar sums of squares of Littlewood-Richardson coefficients. - M. F. Hasler, Jan 20 2020

			For n=3 the s_lambda^2 summed over all partitions of n and decomposed into a sum of Schur functions yields
    s(6) + 2 s(3,3) + 2 s(4,2) + s(5,1) + 2 s(2,2,2) + 2 s(3,2,1) + s(4,1,1)
    + 2 s(2,2,1,1) + s(3,1,1,1) + s(2,1,1,1,1) + s(1,1,1,1,1,1),
  and the sum of the squares of the coefficients {1, 2, 2, 1, 2, 2, 1, 2, 1, 1, 1} gives a(3) = 26.

Seiichi Manyama, Table of n, a(n) for n = 0..1000
J. B. Geloun and S. Ramgoolam, Counting Tensor Model Observables and Branched Covers of the 2-Sphere, arXiv preprint arXiv:1307.6490 [hep-th], 2013.
Eric Weisstein's World of Mathematics, Partition.
Wikipedia, Littlewood-Richardson rule, as of Dec 18 2018.
Wikipedia, Schur polynomial, as of Jan 13 2020.

Cf. A001868.
List of partitions: A036037, A080577, A181317, A330370.
Cf. A070933 (Sum_{lambda,mu,nu} (c^{lambda}_{mu,nu})^2, |mu| = |nu| = n).
Cf. A003040 (maximum number of standard tableaux of the Ferrers diagrams of the partitions of n).

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i=1,
      binomial(n+n, n), add(b(j, 1)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..33);  # Alois P. Heinz, Aug 24 2019

Table[Tr[(Apply[List,
  Sum[Tr[s @@@ LRRule[\[Lambda], \[Lambda]]],
   {\[Lambda], Partitions[n]}]] /. s[] -> 1)^2], {n, 1, 10}];
(* with 'LRRule' defined in http://users.telenet.be/Wouter.Meeussen/ToolBox.nb - Wouter Meeussen, Jan 19 2020 *)
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i == 1, Binomial[n+n, n],
     Sum[b[j, 1]*b[n - i*j, i-1], {j, 0, n/i}]]];
a[n_] := b[n, n];
Table[a[n], {n, 0, 33}] (* Jean-François Alcover, Jan 02 2022, after Alois P. Heinz *)

A067855_upto(N)=Vec(1/sqrt(prod(i=1,N-1,1-4*'x^i+O('x^N)))) \\ M. F. Hasler, Jan 23 2020

G.f.: 1/sqrt(Product_{i >= 1} (1 - 4*x^i)).
Euler transform of A001868(n)/2. a(n) = Sum_{pi} Product_{m=1..n} binomial(2*p(m), p(m)), where pi runs through all nonnegative solutions of p(1) + 2*p(2) + ... + n*p(n)=n. - Vladeta Jovovic, Mar 25 2006
a(n) ~ 2^(2*n) / sqrt(c*Pi*n), where c = QPochhammer[1/4] = 0.688537537120339... - Vaclav Kotesovec, Apr 22 2018
By definition, a(n) = Sum_{mu |- 2n} c_mu^2 where Sum_{lambda |- n} s_lambda^2 = Sum_{mu |- 2n} c_mu s_mu, where s_lambda are the Schur polynomials (symmetric in 2n variables) and the sums run over all partitions of n resp. 2n. - M. F. Hasler, Jan 19 2020

More terms from Vladeta Jovovic, Mar 25 2006

Name edited by M. F. Hasler following observations by Wouter Meeussen, Jan 17 2020

A210109 Number of 3-divided binary sequences (or words) of length n.

Original entry on oeis.org

0, 0, 0, 2, 7, 23, 54, 132, 290, 634, 1342, 2834, 5868, 12140, 24899, 50929, 103735, 210901, 427623, 865910, 1750505, 3535098, 7131321, 14374647, 28952661, 58280123, 117248217, 235770302, 473897980, 952183214, 1912535827, 3840345963, 7709282937, 15472242645, 31045402788, 62280978042

Offset: 1

N. J. A. Sloane, Mar 17 2012

nonn

A binary sequence (or word) W of length n is 3-divided if it can be written as a concatenation W = XYZ such that XYZ is strictly earlier in lexicographic order than any of the five permutations  XZY, ZYX, YXZ, YZX, ZXY.
More generally, fix an alphabet A = {0,1,2,...,a-1}.
Define lexicographic order on words over A in the obvious way: for single letters, i < j is the natural order; for words U, V, we set U < V iff u_i < v_i at the first place where they differ; also U < UV if V is nonempty, etc.
Then a word U over A is "k-divided over A" if it can be written as U = X_1 X_2 ... X_k in such a way that X is strictly less in lexicographic order than X_pi_1 X_pi_2 ... X_pi_k for all nontrivial permutations pi of [1..k].
All 2^n binary words are 1-divided. For 2-divided words see A209970.
"k-divisible" would sound better to me than "k-divided", but I have followed Lothaire and Pirillo-Varricchio in using the latter term. Neither reference gives this sequence.

			The two 3-divisible binary words of length 4 and the seven of length 5 are as follows. The periods indicate the division w = x.y.z. For example, 0.01.1 is 3-divided since 0011 < all of 0101, 1010, 0101, 1001, 0110.
0.01.1
0.10.1
0.001.1
0.010.1
0.01.10
0.01.11
0.100.1
0.10.11
0.110.1

M. Lothaire, Combinatorics on words. A collective work by Dominique Perrin, Jean Berstel, Christian Choffrut, Robert Cori, Dominique Foata, Jean Eric Pin, Guiseppe Pirillo, Christophe Reutenauer, Marcel-P. Schützenberger, Jacques Sakarovitch and Imre Simon. With a foreword by Roger Lyndon. Edited and with a preface by Perrin. Encyclopedia of Mathematics and its Applications, 17. Addison-Wesley Publishing Co., Reading, Mass., 1983. xix+238 pp. ISBN: 0-201-13516-7, MR0675953 (84g:05002). See p. 144.

Michael S. Branicky, Python program using bitwise operations
Giuseppe Pirillo and Stefano Varricchio, Some combinatorial properties of infinite words and applications to semigroup theory. Proceedings of the 5th Conference on Formal Power Series and Algebraic Combinatorics (Florence, 1993). Discrete Math. 153 (1996), no. 1-3, pages 239-251, MR1394958 (98f:05018).

Number of k-divided words of length n over alphabet of size A:
A=2, k=2,3,4,5: A209970 (and A209919, A000031, A001037), A210109 (and A210145), A210321, A210322;
A=3, k=2,3,4,5: A210323 (and A001867, A027376), A210324, A210325, A210326;
A=4, k=2,3,4: A210424 (and A001868, A027377), A210425, A210426.

# see link for faster, bit-based version
from itertools import product
def is3div(b):
    for i in range(1, len(b)-1):
        for j in range(i+1, len(b)):
            X, Y, Z = b[:i], b[i:j], b[j:]
            if all(b < bp for bp in [Z+Y+X, Z+X+Y, Y+Z+X, Y+X+Z, X+Z+Y]):
                return True
    return False
def a(n): return sum(is3div("".join(b)) for b in product("01", repeat=n))
print([a(n) for n in range(1, 16)]) # Michael S. Branicky, Aug 27 2021

Is there a formula or recurrence?

Values confirmed and a(30)-a(31) by David Applegate, Mar 19 2012

a(32)-a(36) from Michael S. Branicky, Aug 27 2021

A032275 Number of bracelets (turnover necklaces) of n beads of 4 colors.

Original entry on oeis.org

4, 10, 20, 55, 136, 430, 1300, 4435, 15084, 53764, 192700, 704370, 2589304, 9608050, 35824240, 134301715, 505421344, 1909209550, 7234153420, 27489127708, 104717491064, 399827748310, 1529763696820

Offset: 1

Christian G. Bower

nonn

			For n=2, the ten bracelets are AA, AB, AC, AD, BB, BC, BD, CC, CD, and DD. - _Robert A. Russell_, Sep 24 2018

C. G. Bower, Transforms (2)
F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc.
F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc. [Cached copy, with permission, pdf format only]
M. Taniguchi, H. Du, and J. S. Lindsey, Enumeration of virtual libraries of combinatorial modular macrocyclic (bracelet, necklace) architectures and their linear counterparts, Journal of Chemical Information and Modeling, 53 (2013), 2203-2216.
Index entries for sequences related to bracelets

Column 4 of A051137.

mx=40;CoefficientList[Series[(1-Sum[ EulerPhi[n]*Log[1-4*x^n]/n,{n,mx}]+(1+4 x+6 x^2)/(1-4 x^2))/2,{x,0,mx}],x] (* Herbert Kociemba, Nov 02 2016 *)
k=4; Table[DivisorSum[n, EulerPhi[#] k^(n/#) &]/(2n) + (k^Floor[(n+1)/2] + k^Ceiling[(n+1)/2])/4, {n, 1, 30}] (* Robert A. Russell, Sep 24 2018 *)

"DIK" (bracelet, indistinct, unlabeled) transform of 4, 0, 0, 0, ...
Equals (A001868(n) + A056486(n)) / 2 = A001868(n) - A278640(n) = A278640(n) + A056486(n), for n>=1.
a(n) = A081720(n,4), n >= 4. - Wolfdieter Lang, Jun 03 2012
G.f.: (1 - Sum_{n>=1} phi(n)*log(1 - 4*x^n)/n + (1+4*x+6*x^2)/(1-4*x^2))/2. - Herbert Kociemba, Nov 02 2016
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=4 is the maximum number of colors. - Robert A. Russell, Sep 24 2018
a(n) = (k^floor((n+1)/2) + k^ceiling((n+1)/2))/4 + (1/(2*n))*Sum_{i=1..n} k^gcd(n,i), where k=4 is the maximum number of colors. (See A075195 formulas.) - Richard L. Ollerton, May 04 2021

A056292 Number of n-bead necklace structures using a maximum of four different colored beads.

Original entry on oeis.org

1, 2, 3, 7, 11, 39, 103, 367, 1235, 4439, 15935, 58509, 215251, 799697, 2983217, 11187567, 42109451, 159082753, 602809327, 2290684251, 8726308317, 33318661277, 127479700199, 488672302909, 1876500180291, 7217308815887, 27799998949873, 107228568948547

Offset: 1

Marks R. Nester

nonn

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

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]

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

Cf. A000013, A001868.

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, 4]; Table[Print[an = a[n]]; an, {n, 1, 25}] (* Jean-François Alcover, Oct 04 2013, after N. J. A. Sloane's Maple code *)
(* This 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]];
Table[SeriesCoefficient[DivisorSum[n, EulerPhi[#] Adn[#, n/#] &]
/(n (1 - x)), {x, 0, 4}], {n, 1, 40}] (* Robert A. Russell, Feb 24 2018 *)
From Robert A. Russell, May 29 2018: (Start)
Table[(1/n) DivisorSum[n, EulerPhi[#] Which[Divisible[#,12], 4 StirlingS2[n/#+3,4] - 24 StirlingS2[n/#+2,4] + 44 StirlingS2[n/#+1,4] - 24 StirlingS2[n/#,4], Divisible[#,6], 3 StirlingS2[n/#+3,4] - 18 StirlingS2[n/#+2,4] + 33 StirlingS2[n/#+1,4] - 18 StirlingS2[n/#,4], Divisible[#,4], 3 StirlingS2[n/#+3,4] - 19 StirlingS2[n/#+2,4] + 38 StirlingS2[n/#+1,4] - 24 StirlingS2[n/#,4], Divisible[#,3], 2 StirlingS2[n/#+3,4] - 13 StirlingS2[n/#+2,4] + 26 StirlingS2[n/#+1,4] - 15 StirlingS2[n/#,4], Divisible[#,2], 2 StirlingS2[n/#+3,4] - 13 StirlingS2[n/#+2,4] + 27 StirlingS2[n/#+1,4] - 18 StirlingS2[n/#,4], True, StirlingS2[n/#+3,4] - 8 StirlingS2[n/#+2,4] + 20 StirlingS2[n/#+1,4] - 15 StirlingS2[n/#,4]] &],{n, 1, 40}]
mx = 40; Drop[CoefficientList[Series[1 - Sum[(EulerPhi[d] / d) Which[
  Divisible[d, 12], Log[1 - 4x^d], Divisible[d, 6],
  3 Log[1 - 4x^d] / 4, Divisible[d, 4] ,
  (2 Log[1 - 4x^d] + Log[1 - x^d]) / 3, Divisible[d, 3],
  (3 Log[1 - 4x^d] + 2 Log[1 - 2x^d]) / 8,
  Divisible[d, 2], (5 Log[1 - 4x^d] + 4 Log[1 - x^d]) / 12,
  True, (Log[1 - 4x^d] + 6 Log[1 - 2x^d] + 8 Log[1 - x^d]) / 24], {d, 1, mx}], {x, 0, mx}], x], 1]
(End)

Use de Bruijn's generalization of Polya's enumeration theorem as discussed in reference.
From Robert A. Russell, May 29 2018: (Start)
a(n) = (1/n) * Sum_{d|n} phi(d) * ([d==0 mod 12] * (4*S2(n/d+3, 4) - 24*S2(n/d+2, 4) + 44*S2(n/d+1, 4) - 24*S2(n/d, 4)) + [d==6 mod 12] * (3*S2(n/d+3, 4) - 18*S2(n/d+2, 4) + 33*S2(n/d+1, 4) - 18*S2(n/d, 4)) + [d==4 mod 12 | d==8 mod 12] * (3*S2(n/d+3, 4) - 19*S2(n/d+2, 4) + 38*S2(n/d+1, 4) - 24*S2(n/d, 4)) + [d==3 mod 12 | d=9 mod 12] * (2*S2(n/d+3, 4) - 13*S2(n/d+2, 4) + 26*S2(n/d+1, 4) - 15*S2(n/d, 4)) + [d==2 mod 12 | d=10 mod 12] * (2*S2(n/d+3, 4) - 13*S2(n/d+2, 4) + 27*S2[n/d+1,4) - 18*S2(n/d, 4)) + [d mod 12 in {1,5,7,11}] * (S2(n/d+3, 4) - 8*S2(n/d+2, 4) + 20*S2(n/d+1, 4) - 15*S2(n/d, 4))), where S2(n, k) is the Stirling subset number, A008277.
G.f.: 1 - Sum_{d>0} (phi(d) / d) * ([d==0 mod 12] * log(1-4x^d) + [d==6 mod 12] * 3*log(1-4x^d) / 4 + [d==4 mod 12 | d==8 mod 12] * (2*log(1-4x^d) + log(1-x^d)) / 3 + [d==3 mod 12 | d=9 mod 12] * (3*log(1-4x^d) + 2*log(1-2x^d)) / 8 + [d==2 mod 12 | d=10 mod 12] * (5*log(1-4x^d) + 4*log(1-x^d)) / 12 + [d mod 12 in {1,5,7,11}] * (log(1-4x^d) + 6*log(1-2x^d) + 8*log(1-x^d)) / 24).
(End)

A054719 Number of 4-ary sequences with primitive period n.

Original entry on oeis.org

1, 4, 12, 60, 240, 1020, 4020, 16380, 65280, 262080, 1047540, 4194300, 16772880, 67108860, 268419060, 1073740740, 4294901760, 17179869180, 68719210560, 274877906940, 1099510578960, 4398046494660, 17592181850100, 70368744177660, 281474959868160

Offset: 0

N. J. A. Sloane, Apr 20 2000

nonn

Equivalently, output sequences with primitive period n from a simple cycling shift register.

Seiichi Manyama, Table of n, a(n) for n = 0..1660 (terms 0..500 from Alois P. Heinz)
E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.

Cf. A001868, A027377.

Column k=4 of A143324.

A054719 := proc(n) local d,s; if n = 0 then RETURN(1); else s := 0; for d in divisors(n) do s := s+mobius(d)*4^(n/d); od; RETURN(s); fi; end;

a[0] = 1; a[n_] := Sum[MoebiusMu[d]*4^(n/d), {d, Divisors[n]}]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Mar 11 2014 *)

a(n) = Sum_{d|n} mu(d)*4^(n/d).
a(n) = n * A027377(n), n>0.
G.f.: 1 + 4 * Sum_{k>=1} mu(k) * x^k / (1 - 4*x^k). - Ilya Gutkovskiy, Apr 14 2021

A184277 Table read by antidiagonals: T(n,k) = number of distinct n X k toroidal 0..3 arrays.

Original entry on oeis.org

4, 10, 10, 24, 76, 24, 70, 700, 700, 70, 208, 8296, 29184, 8296, 208, 700, 104968, 1398500, 1398500, 104968, 700, 2344, 1399176, 71582944, 268447936, 71582944, 1399176, 2344, 8230, 19175140, 3817765120, 54975633976, 54975633976

Offset: 1

R. H. Hardin, Jan 10 2011

			Table starts
      4        10           24             70            208            700
     10        76          700           8296         104968        1399176
     24       700        29184        1398500       71582944     3817765120
     70      8296      1398500      268447936    54975633976 11728126132976
    208    104968     71582944    54975633976 45035996274688
    700   1399176   3817765120 11728126132976
   2344  19175140 209430787824
   8230 268447816
  29144

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 48 terms from R. H. Hardin)
S. N. Ethier, Counting toroidal binary arrays, arXiv:1301.2352v1 [math.CO], Jan 10, 2013.
S. N. Ethier and Jiyeon Lee, Counting toroidal binary arrays, II, arXiv:1502.03792v1 [math.CO], Feb 12, 2015.
Veronika Irvine, Lace Tessellations: A mathematical model for bobbin lace and an exhaustive combinatorial search for patterns, PhD Dissertation, University of Victoria, 2016.
Peter Kagey and William Keehn, Counting tilings of the n X m grid, cylinder, and torus, arXiv:2311.13072 [math.CO], 2023. See pp. 3, 42.

Columns 1-5 are A001868, A184273, A184274, A184275, A184276.
Main diagonal is A184272.
Cf. A184271, A184284.

T[n_, k_] := (1/(n*k))*Sum[Sum[EulerPhi[c]*EulerPhi[d]*4^(n*k/LCM[c, d]), {d, Divisors[k]}], {c, Divisors[n]}];
Table[T[n-k+1, k], {n, 1, 9}, {k, 1, n}] // Flatten (* Jean-François Alcover, Oct 31 2017, after Andrew Howroyd *)

T(n, k) = (1/(n*k)) * sumdiv(n, c, sumdiv(k, d, eulerphi(c) * eulerphi(d) * 4^(n*k/lcm(c,d)))); \\ Andrew Howroyd, Sep 27 2017

T(n,k) = (1/(n*k)) * Sum_{c|n} Sum_{d|k} phi(c) * phi(d) * 4^(n*k/lcm(c,d)). - Andrew Howroyd, Sep 27 2017

A056284 Number of n-bead necklaces with exactly four different colored beads.

Original entry on oeis.org

0, 0, 0, 6, 48, 260, 1200, 5106, 20720, 81876, 318000, 1223136, 4675440, 17815020, 67769552, 257700906, 980240880, 3731753180, 14222737200, 54278580036, 207438938000, 793940475900, 3043140078000, 11681057249536, 44900438149296, 172824331826580, 666070256489680

Offset: 1

Marks R. Nester

nonn

Turning over the necklace is not allowed.

			For n=4, the six necklaces are ABCD, ABDC, ACBD, ACDB, ADBC and ADCB.

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. A001868.

Column k=4 of A087854.

k=4; Table[k!DivisorSum[n,EulerPhi[#]StirlingS2[n/#,k]&]/n,{n,1,30}] (* Robert A. Russell, Sep 26 2018 *)

a(n) = my(k=4);(k!/n)*sumdiv(n, d, eulerphi(d)*stirling(n/d,k,2)); \\ Michel Marcus, Sep 27 2018

a(n) = A001868(n) - 4*A001867(n) + 6*A000031(n) - 4.
From Robert A. Russell, Sep 26 2018: (Start)
a(n) = (k!/n) Sum_{d|n} phi(d) S2(n/d,k), where k=4 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=4 is the number of colors. (End)

A056285 Number of n-bead necklaces with exactly five different colored beads.

Original entry on oeis.org

0, 0, 0, 0, 24, 300, 2400, 15750, 92680, 510312, 2691600, 13794150, 69309240, 343501500, 1686135376, 8221437000, 39901776360, 193054016840, 932142850800, 4495236798162, 21664357535320, 104388120866100, 503044634004000, 2425003924383900, 11696087875731624

Offset: 1

Marks R. Nester

nonn

Turning over the necklace is not allowed.

			For n=5, the 24 necklaces are A followed by the 24 permutations of BCDE.

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, A001868, A001869, A008277.

Column k=5 of A087854.

k=5; Table[k!DivisorSum[n,EulerPhi[#]StirlingS2[n/#,k]&]/n,{n,1,30}] (* Robert A. Russell, Sep 26 2018 *)

a(n) = my(k=5); k!*sumdiv(n, d, eulerphi(d)*stirling(n/d, k, 2))/n; \\ Michel Marcus, Sep 27 2018

a(n) = A001869(n) - 5*A001868(n) + 10*A001867(n) - 10*A000031(n) + 5.
From Robert A. Russell, Sep 26 2018: (Start)
a(n) = (k!/n) Sum_{d|n} phi(d) S2(n/d,k), where k=5 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=5 is the number of colors. (End)

A054611 a(n) = Sum_{d|n} phi(d)*4^(n/d).

Original entry on oeis.org

0, 4, 20, 72, 280, 1040, 4200, 16408, 65840, 262296, 1049680, 4194344, 16782000, 67108912, 268451960, 1073744160, 4295033440, 17179869248, 68719747320, 274877907016, 1099512679520, 4398046544304, 17592190238920, 70368744177752

Offset: 0

N. J. A. Sloane, Apr 16 2000

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1660
T. Pisanski, D. Schattschneider and B. Servatius, Applying Burnside's lemma to a one-dimensional Escher problem, Math. Mag., 79 (2006), 167-180. See V(n).

Column k=4 of A185651.
Row n=4 of A054619.
Cf. A001868.

A054611:=proc(n) local k, t1; t1:=0; for k in divisors(n) do t1 := t1+phi(k)*4^(n/k); od: t1; end;

a(n) = if(n==0, 0, sumdiv(n, d, eulerphi(d)*4^(n/d))); \\ Michel Marcus, Sep 19 2017

a(n) = n * A001868(n).

a(n) = Sum_{k=1..n} 4^gcd(n,k). - Ilya Gutkovskiy, Apr 16 2021

cp's OEIS Frontend

A075195 Jablonski table T(n,k) read by antidiagonals: T(n,k) = number of necklaces with n beads of k colors.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

Extensions

A067855 Square of the Euclidean length of the vector of Littlewood-Richardson coefficients of Sum_{lambda |- n} s_lambda^2, where s_lambda are the symmetric Schur functions and the sum runs over all partitions lambda of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A210109 Number of 3-divided binary sequences (or words) of length n.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Python

Formula

Extensions

A032275 Number of bracelets (turnover necklaces) of n beads of 4 colors.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A056292 Number of n-bead necklace structures using a maximum of four different colored beads.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

Formula

A054719 Number of 4-ary sequences with primitive period n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A184277 Table read by antidiagonals: T(n,k) = number of distinct n X k toroidal 0..3 arrays.

Views

Author

Keywords

Examples