A053656 -id:A053656 - cp's OEIS frontend

A256216 a(n) = A053656(n) - A000011(n).

0, 0, 0, 0, 0, 1, 1, 4, 7, 18, 31, 70, 126, 261, 484, 960, 1800, 3515, 6643, 12852, 24458, 47151, 90157, 173744, 333498, 643230, 1238671, 2392650, 4620006, 8939676, 17302033, 33538048, 65042526, 126289800, 245361172

Offset: 1

N. J. A. Sloane, Mar 26 2015

nonn

Counts contiguously substituted cycloalkane polyols (CSCPs).
From Ed Wynn and Andrew Howroyd, May 22 2021: (Start)
Consider a bracelet of n beads, each colored blue on one side and red on the other. Turning the bracelet over has the effect of simultaneously swapping the colors and reversing the order of the beads. For example, rrbbrb when turned over becomes rbrrbb. The total number of such bracelets is counted by A053656(n).
Swapping the colors is equivalent to reversing the order of the beads. For example, rrbbrb becomes bbrrbr which when turned over is brbbrr. A bracelet may or may not be the same as its reversal (or complement). The case of equality is counted by A256217(n) and the remainder can be divided into "chiral" pairs which are the reverse of each other and counted by this sequence. a(n) is then the number of pairs of two-colored n-bead bracelets that are equal under reversal but unequal up to rotation and turning over.
In chemical terms, these pairs are called "enantiomeric pairs". The example of rrbbrb corresponds to a pair of "chiral" chemical molecules: L-chiro-inositol and R-chiro-inositol.
a(n) is also half the number of nonisomorphic orientations of the n-cycle graph which are not self-converse. Again the self-converse orientations are counted by A256217(n) and the total by A053656(n).
(End)

			From _Ed Wynn_ and _Andrew Howroyd_, May 22 2021: (Start)
The a(6) = 1 pair of bracelets are rrbbrb and its complement bbrrbr. These two are not the same under simultaneous reversal and swapping the colors (rrbbrb is equivalent to rbrrbb which is not the same as bbrrbr by rotation).
Replacing r with ->- and b with -<- gives two distinct orientations of the cycle:
     ->-.->-.-<-.-<-.->-.-<-   :   ->-.-<-.->-.->-.-<-.-<-
    |                       |  :  |                       |
     -----------.-----------   :   -----------.-----------
These two might be written in shorthand as >><<>< and <<>><>.
The a(8) = 4 pairs of bracelets are rrrrbrbb, rrrbrrbb, rrrbrbbb, rrbrbrbb and their complements.
(End)

Seiichi Manyama, Table of n, a(n) for n = 1..3335
D. Bundala, M. Codish, L. Cruz-Filipe et al., Optimal-Depth Sorting Networks, arXiv preprint arXiv:1412.5302 [cs.DS], 2014. See Table 4 and associated comments.
Shinsaku Fujita, alpha-beta Itemized Enumeration of Inositol Derivatives and m-Gonal Homologs by Extending Fujita's Proligand Method, Bull. Chem. Soc. Jpn. 2017, 90, 343-366 | doi:10.1246/bcsj.20160369. See Table 8.
A. Yajima, How to calculate the number of stereoisomers of inositol-homologs, Bull. Chem. Soc. Jpn. 2014, 87, 1260-1264; doi:10.1246/bcsj.20140204. See Table 2 (E_c).

Cf. A000011, A053656, A256217, A066314.

The 8 sequences in Table 8 of Fujita (2017) are A053656, A000011, A256216, A256217, A123045, A283846, A283847, A283848.

Table[Total[EulerPhi[#] 2^(n/#) & /@ Divisors[n]]/(2 n) + 2^(n/2 - 2) (1 - Mod[n, 2]) - If[n < 1, Boole[n == 0], 2^Quotient[n, 2]/2 + DivisorSum[n, EulerPhi[2 #] 2^(n/#) &]/(4 n)], {n, 35}] (* Michael De Vlieger, Sep 05 2015, after Jean-François Alcover at A053656 and Michael Somos at A000011 *)

a(n) = A053656(n) - A000011(n).

A053656(n) = 2*a(n) + A256217(n). - Andrew Howroyd and Ed Wynn, Jun 15 2021

A000011 Number of n-bead necklaces (turning over is allowed) where complements are equivalent.

Original entry on oeis.org

1, 1, 2, 2, 4, 4, 8, 9, 18, 23, 44, 63, 122, 190, 362, 612, 1162, 2056, 3914, 7155, 13648, 25482, 48734, 92205, 176906, 337594, 649532, 1246863, 2405236, 4636390, 8964800, 17334801, 33588234, 65108062, 126390032, 245492244, 477353376, 928772650, 1808676326, 3524337980

Offset: 0

N. J. A. Sloane

a(n) is also the number of minimal fibrations of a bidirectional n-cycle over the 2-bouquet up to precompositions with automorphisms of the n-cycle and postcomposition with automorphisms of the 2-bouquet. (Boldi et al.) - Sebastiano Vigna, Jan 08 2018

For n >= 3, also the number of distinct planar embeddings of the n-sunlet graph. - Eric W. Weisstein, May 21 2024

			From Jason Orendorff (jason.orendorff(AT)gmail.com), Jan 09 2009: (Start)
The binary bracelets for small n are:
  n: bracelets
  0: (the empty bracelet)
  1: 0
  2: 00, 01
  3: 000, 001
  4: 0000, 0001, 0011, 0101
  5: 00000, 00001, 00011, 00101
  6: 000000, 000001, 000011, 000101, 000111, 001001, 001011, 010101
(End)

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..3335 (first 201 terms from T. D. Noe)
Joerg Arndt, Matters Computational (The Fxtbook)
Paolo Boldi and Sebastiano Vigna, Fibrations of Graphs, Discrete Math., 243 (2002), 21-66.
H. Bottomley, Initial terms of A000011 and A000013
Aharon Davidson, From Planck Area to Graph Theory: Topologically Distinct Black Hole Microstates, arXiv:1907.03090 [gr-qc], 2019.
Daniel T. Eatough and Keith A. Seffen, Calculating the Fold Angles of Any Vertex Roof Using a Spherical Image Technique, J. Mechanisms Robotics (2020) Vol. 12, No. 3, 031004.
N. J. Fine, Classes of periodic sequences, Illinois J. Math., 2 (1958), 285-302.
Shinsaku Fujita, alpha-beta Itemized Enumeration of Inositol Derivatives and m-Gonal Homologs by Extending Fujita's Proligand Method, Bull. Chem. Soc. Jpn. 2017, 90, 343-366; doi:10.1246/bcsj.20160369. See Table 8.
E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.
W. D. Hoskins and Anne Penfold Street, Twills on a given number of harnesses, J. Austral. Math. Soc. Ser. A 33 (1982), no. 1, 1-15.
W. D. Hoskins and A. P. Street, Twills on a given number of harnesses, J. Austral. Math. Soc. (Series A), 33 (1982), 1-15. (Annotated scanned copy)
Yi Hu, Numerical Transfer Matrix Method of Next-nearest-neighbor Ising Models, Master's Thesis, Duke Univ. (2021).
Yi Hu and Patrick Charbonneau, Numerical transfer matrix study of frustrated next-nearest-neighbor Ising models on square lattices, arXiv:2106.08442 [cond-mat.stat-mech], 2021.
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]
A. P. Street, Letter to N. J. A. Sloane, N.D.
Zhe Sun, T. Suenaga, P. Sarkar, S. Sato, M. Kotani, and H. Isobe, Stereoisomerism, crystal structures, and dynamics of belt-shaped cyclonaphthylenes, Proc. Nat. Acad. Sci. USA, vol. 113 no. 29, pp. 8109-8114, doi: 10.1073/pnas.1606530113.
Eric Weisstein's World of Mathematics, Planar Embedding.
Eric Weisstein's World of Mathematics, Sunlet Graph.
A. Yajima, How to calculate the number of stereoisomers of inositol-homologs, Bull. Chem. Soc. Jpn. 2014, 87, 1260-1264; doi:10.1246/bcsj.20140204. See Tables 1 and 2 (and text).
Index entries for sequences related to necklaces
Index entries for sequences related to bracelets

Column 2 of A320748.
Cf. A000013. Bisections give A000117 and A092668.
The 8 sequences in Table 8 of Fujita (2017) are A053656, A000011, A256216, A256217, A123045, A283846, A283847, A283848.

with(numtheory): A000011 := proc(n) local s,d; if n = 0 then RETURN(1) else s := 2^(floor(n/2)); for d in divisors(n) do s := s+(phi(2*d)*2^(n/d))/(2*n); od; RETURN(s/2); fi; end;

a[n_] := Fold[ #1 + EulerPhi[2#2]2^(n/#2)/(2n) &, 2^Floor[n/2], Divisors[n]]/2
a[ n_] := If[ n < 1, Boole[n == 0], 2^Quotient[n, 2] / 2 + DivisorSum[ n, EulerPhi[2 #] 2^(n/#) &] / (4 n)]; (* Michael Somos, Dec 19 2014 *)

{a(n) = if( n<1, n==0, 2^(n\2) / 2 + sumdiv(n, k, eulerphi(2*k) * 2^(n/k)) / (4*n))}; /* Michael Somos, Jun 03 2002 */

a(n) = (A000013(n) + 2^floor(n/2))/2.

Better description from Christian G. Bower

More terms from David W. Wilson, Jan 13 2000

A123045 Number of frieze patterns of length n under a certain group (see Pisanski et al. for precise definition).

Original entry on oeis.org

0, 2, 6, 12, 39, 104, 366, 1172, 4179, 14572, 52740, 190652, 700274, 2581112, 9591666, 35791472, 134236179, 505290272, 1908947406, 7233629132, 27488079132, 104715393912, 399823554006, 1529755308212, 5864066561554, 22517998136936, 86607703209516

Offset: 0

N. J. A. Sloane, Nov 11 2006

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Shinsaku Fujita, alpha-beta Itemized Enumeration of Inositol Derivatives and m-Gonal Homologs by Extending Fujita's Proligand Method, Bull. Chem. Soc. Jpn. 2017, 90, 343-366; doi:10.1246/bcsj.20160369. See Table 8.
T. Pisanski, D. Schattschneider and B. Servatius, Applying Burnside's lemma to a one-dimensional Escher problem, Math. Mag., 79 (2006), 167-180. See F(n).

Cf. A054611, A018215.

The 8 sequences in Table 8 of Fujita (2017) are A053656, A000011, A256216, A256217, A123045, A283846, A283847, A283848.

with(numtheory):
V:=proc(n) local k, t1; t1:=0; for k in divisors(n) do t1 := t1+phi(k)*4^(n/k); od: t1; end; # A054611
H:=n-> if n mod 2 = 0 then (n/2)*4^(n/2); else 0; fi; # this is A018215 interleaved with 0's
A123045:=n-> `if`(n=0,0, (V(n)+H(n))/(2*n));

V[n_] := Module[{t1 = 0}, Do[t1 = t1 + EulerPhi[k] 4^(n/k), {k, Divisors[n]}]; t1];
H[n_] := If[Mod[n, 2] == 0, (n/2) 4^(n/2), 0];
a[n_] := If[n == 0, 0, (V[n] + H[n])/(2n)];
a /@ Range[0, 26] (* Jean-François Alcover, Mar 20 2020, from Maple *)

See Maple program.

A256217 a(n) = A000011(n) - A256216(n).

Original entry on oeis.org

1, 2, 2, 4, 4, 7, 8, 14, 16, 26, 32, 52, 64, 101, 128, 202, 256, 399, 512, 796, 1024, 1583, 2048, 3162, 4096, 6302, 8192, 12586, 16384, 25124, 32768, 50186, 65536, 100232, 131072, 200266, 262144, 400115, 524288, 799568, 1048576, 1597834, 2097152, 3193438, 4194304, 6382637, 8388608, 12757770, 16777216, 25501370

Offset: 1

N. J. A. Sloane, Mar 26 2015

nonn

Counts contiguously substituted cycloalkane polyols (CSCPs).

Consider a bracelet of n beads, each colored blue on one side and red on the other -- each bead changes color when the bracelet is turned over. Reversal is then the same as swapping the colors. a(n) is the number of colorings that are invariant under reversal, up to rotation and turning over. - Ed Wynn, May 22 2021

			The a(7) = 8 bracelets are rrrrrrr, rrrrrrb, rrrrrbb, rrrrbbb, rrrrbrb, rrrbrrb, rrbbrrb, rrbrbrb. - _Ed Wynn_, May 22 2021
a(12) = A056503(12) + 1. The extra bracelet is rrrbrrbbbrbb. - _Andrew Howroyd_, Jun 25 2021

Seiichi Manyama, Table of n, a(n) for n = 1..6644
Shinsaku Fujita, alpha-beta Itemized Enumeration of Inositol Derivatives and m-Gonal Homologs by Extending Fujita's Proligand Method, Bull. Chem. Soc. Jpn. 2017, 90, 343-366; doi:10.1246/bcsj.20160369. See Table 8.
A. Yajima, How to calculate the number of stereoisomers of inositol-homologs, Bull. Chem. Soc. Jpn. 2014, 87, 1260-1264 | doi:10.1246/bcsj.20140204. See Table 2 (M_c).

Cf. A000011, A053656, A056503, A066314, A256216.

The 8 sequences in Table 8 of Fujita (2017) are A053656, A000011, A256216, A256217, A123045, A283846, A283847, A283848.

a(n) = A056503(n) for odd n. - Andrew Howroyd, Jun 14 2021

A283846 Number of n-gonal inositol homologs with 2 kinds of achiral proligands.

Original entry on oeis.org

2, 6, 10, 31, 68, 226, 650, 2259, 7542, 27036, 96350, 352786, 1294652, 4806366, 17912120, 67160083, 252710672, 954641186, 3617076710, 13744708060, 52358745532, 199914446106, 764881848410, 2932043941394, 11259015845684, 43303894193076, 166800053312630

Offset: 1

N. J. A. Sloane, Apr 01 2017

nonn

Counts A032275 up to paired color permutation (equivalent to full color permutation on the 2-tuples of two subcolors, e.g., convert quaternary beads 0 1 2 3 to dibit beads 00 01 10 11). - Travis Scott, Jan 09 2023

Robert Israel, Table of n, a(n) for n = 1..1665
Shinsaku Fujita, alpha-beta Itemized Enumeration of Inositol Derivatives and m-Gonal Homologs by Extending Fujita's Proligand Method, Bull. Chem. Soc. Jpn. 2017, 90, 343-366; doi:10.1246/bcsj.20160369. See Table 8.
Yi Hu, Numerical Transfer Matrix Method of Next-nearest-neighbor Ising Models, Master's Thesis, Duke Univ. (2021).
Yi Hu and Patrick Charbonneau, Numerical transfer matrix study of frustrated next-nearest-neighbor Ising models on square lattices, arXiv:2106.08442 [cond-mat.stat-mech], 2021.

The 8 sequences in Table 8 of Fujita (2017) are A053656, A000011, A256216, A256217, A123045, A283846, A283847, A283848.

f:=  proc(m) uses numtheory;
  if m::even then 1/(4*m)*add(phi(d)*4^(m/d)*`if`(d::even,2,1), d = divisors(m))
+ 3*2^(m-2)
  else
1/(4*m)*add(phi(d)*4^(m/d),d=divisors(m))+2^(m-1)
  fi
end proc:
map(f, [$1..100]); # Robert Israel, Aug 21 2018

f[m_] := If[EvenQ[m], 1/(4m)*Sum[EulerPhi[d]*4^(m/d)*If[EvenQ[d], 2, 1], {d, Divisors[m]}]+ 3*2^(m-2), 1/(4m)*Sum[EulerPhi[d]*4^(m/d), {d, Divisors[m]}] + 2^(m-1)];
f /@ Range[1, 25] (* Jean-François Alcover, Feb 26 2019, after Robert Israel *)

From Robert Israel, Aug 21 2018 after Fujita (2017), Eq. (99)(set n=2, m=n): (Start)
if n is even, a(n) = (4*n)^(-1)*(Sum_{d|n} phi(d)*4^(n/d) + Sum_{d|n, d even} phi(d)*4^(n/d)) + 3*2^(n-2).
if n is odd, a(n) = 2^(n-1) + (4*n)^(-1)*Sum_{d|n} phi(d)*4^(n/d). (End)

a(1)-a(2) prepended by Travis Scott, Jan 09 2023

A283847 Number of n-gonal inositol homologs with 2 kinds of achiral proligands.

Original entry on oeis.org

2, 8, 36, 140, 522, 1920, 7030, 25704, 94302, 347488, 1286460, 4785300, 17879352, 67076096, 252579600, 954306220, 3616552422, 13743371072, 52356648380, 199909107900, 764873459802, 2932022620160, 11258982291252, 43303809016440, 166799919094902, 643371241120928

Offset: 3

N. J. A. Sloane, Apr 01 2017

nonn

Robert Israel, Table of n, a(n) for n = 3..1665
Shinsaku Fujita, alpha-beta Itemized Enumeration of Inositol Derivatives and m-Gonal Homologs by Extending Fujita's Proligand Method, Bull. Chem. Soc. Jpn. 2017, 90, 343-366; doi:10.1246/bcsj.20160369. See Table 8.

The 8 sequences in Table 8 of Fujita (2017) are A053656, A000011, A256216, A256217, A123045, A283846, A283847, A283848.

f:= proc(n) uses numtheory;
  if n::even then (4*n)^(-1)*add(phi(d)*4^(n/d), d = select(type,divisors(n),odd)) - 2^(n-1)
  else (4*n)^(-1)*add(phi(d)*4^(n/d), d = divisors(n)) - 2^(n-1)
  fi
end proc:
map(f, [$3..50]); # Robert Israel, Aug 23 2018

a[n_] := If[EvenQ[n], (4n)^(-1) Sum[EulerPhi[d] 4^(n/d), {d, Select[ Divisors[n], OddQ]}] - 2^(n-1), (4n)^(-1) Sum[EulerPhi[d] 4^(n/d), {d, Divisors[n]}] - 2^(n-1)];
Table[a[n], {n, 3, 50}] (* Jean-François Alcover, Mar 23 2019, after Robert Israel *)

a(n) = (4*n)^(-1)*(Sum_{d|n, d odd} phi(d)*4^(n/d)) - 2^(n-1). - Robert Israel, Aug 23 2018 after Fujita (2017), Eq. (100) (set n=2, m=n)

A283848 Number of n-gonal inositol homologs with 2 kinds of achiral proligands.

Original entry on oeis.org

8, 23, 32, 86, 128, 339, 512, 1332, 2048, 5298, 8192, 21066, 32768, 83987, 131072, 334966, 524288, 1336988, 2097152, 5338206, 8388608, 21321234, 33554432, 85176636, 134217728, 340338398, 536870912, 1360073016, 2147483648, 5435820051, 8589934592, 21727481616, 34359738368, 86853790498, 137438953472

Offset: 3

N. J. A. Sloane, Apr 01 2017

nonn

Robert Israel, Table of n, a(n) for n = 3..3318
Shinsaku Fujita, alpha-beta Itemized Enumeration of Inositol Derivatives and m-Gonal Homologs by Extending Fujita's Proligand Method, Bull. Chem. Soc. Jpn. 2017, 90, 343-366; doi:10.1246/bcsj.20160369. See Table 8.

The 8 sequences in Table 8 of Fujita (2017) are A053656, A000011, A256216, A256217, A123045, A283846, A283847, A283848.

f:= proc(n) uses numtheory;
  if n::even then (2*n)^(-1)*add(phi(d)*4^(n/d),d=select(type,divisors(n),even))+5*2^(n-2)
  else 2^n
  fi
end proc:
map(f, [$1..40]); # Robert Israel, Aug 23 2018

a[n_] := If[EvenQ[n], (2n)^(-1) Sum[EulerPhi[d] 4^(n/d), {d, Select[ Divisors[n], EvenQ]}] + 5*2^(n-2), 2^n];
Table[a[n], {n, 3, 40}] (* Jean-François Alcover, Mar 23 2019, after Robert Israel *)

a(n) = if (n%2, 2^n, (2*n)^(-1)*sumdiv(n, d, if (!(d%2), eulerphi(d)*4^(n/d))) + 5*2^(n-2)); \\ Michel Marcus, Mar 23 2019

If n is even, a(n) = (2*n)^(-1)*Sum_{d|n, d even} phi(d)*4^(n/d) + 5*2^(n-2). - Robert Israel, Aug 23 2018 after Fujita (2017), Eq. (101) (set n=2, m=n).

If n is odd, a(n) = 2^n. For the even bisection see A284711.

Edited and more terms by Robert Israel, Aug 23 2018

A300190 Number of solutions to 1 +- 2 +- 3 +- ... +- n == 0 (mod n).

Original entry on oeis.org

1, 0, 2, 4, 4, 0, 10, 32, 30, 0, 94, 344, 316, 0, 1096, 4096, 3856, 0, 13798, 52432, 49940, 0, 182362, 699072, 671092, 0, 2485534, 9586984, 9256396, 0, 34636834, 134217728, 130150588, 0, 490853416, 1908874584, 1857283156, 0, 7048151672, 27487790720

Offset: 1

Seiichi Manyama, Feb 28 2018

nonn

Apparently a(2*n + 1) = A053656(2*n + 1) for n >= 0. - Georg Fischer, Mar 26 2019

			Solutions for n = 7:
--------------------------
1 +2 +3 +4 +5 +6 +7 =  28.
1 +2 +3 +4 +5 +6 -7 =  14.
1 +2 -3 +4 -5 -6 +7 =   0.
1 +2 -3 +4 -5 -6 -7 = -14.
1 +2 -3 -4 +5 +6 +7 =  14.
1 +2 -3 -4 +5 +6 -7 =   0.
1 -2 +3 +4 -5 +6 +7 =  14.
1 -2 +3 +4 -5 +6 -7 =   0.
1 -2 -3 -4 -5 +6 +7 =   0.
1 -2 -3 -4 -5 +6 -7 = -14.

Seiichi Manyama, Table of n, a(n) for n = 1..3334 (terms 1..1000 from Alois P. Heinz)

Number of solutions to 1 +- 2^k +- 3^k +- ... +- n^k == 0 (mod n): this sequence (k=1), A300268 (k=2), A300269 (k=3).
Cf. A000016, A026119, A053656, A058377, A063776, A300307, A300328.
Cf. A016825 (4n+2).

b:= proc(n, i, m) option remember; `if`(i=0, `if`(n=0, 1, 0),
      add(b(irem(n+j, m), i-1, m), j=[i, m-i]))
    end:
a:= n-> b(0, n-1, n):
seq(a(n), n=1..60);  # Alois P. Heinz, Mar 01 2018

b[n_, i_, m_] := b[n, i, m] = If[i == 0, If[n == 0, 1, 0], Sum[b[Mod[n + j, m], i - 1, m], {j, {i, m - i}}]];
a[n_] := b[0, n - 1, n];
Array[a, 60] (* Jean-François Alcover, Apr 29 2020, after Alois P. Heinz *)

def A(n)
  ary = [1] + Array.new(n - 1, 0)
  (1..n).each{|i|
    i1 = 2 * i
    a = ary.clone
    (0..n - 1).each{|j| a[(j + i1) % n] += ary[j]}
    ary = a
  }
  ary[(n * (n + 1) / 2) % n] / 2
end
def A300190(n)
  (1..n).map{|i| A(i)}
end
p A300190(100)

a(4*n+1) = A000016(n), a(4*n+2) = 0, a(4*n+3) = A000016(n), a(4*n+4) = 2 * A000016(n) for n > 0.

a(2^n) = 2^A000325(n) for n > 1.

A026119 Bisection of A000016 (also of A000013).

Original entry on oeis.org

1, 2, 4, 10, 30, 94, 316, 1096, 3856, 13798, 49940, 182362, 671092, 2485534, 9256396, 34636834, 130150588, 490853416, 1857283156, 7048151672, 26817356776, 102280151422, 390937468408, 1497207322930, 5744387279818, 22076468764192

Offset: 0

N. J. A. Sloane, Apr 12 2000

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1666

Cf. A000013, A000016, A000116.

Bisection of A053634 and A053656.

a(n) = sumdiv(2*n+1, d, eulerphi(d)*2^((2*n+1)/d)) / (4*n+2); \\ Michel Marcus, Sep 11 2013

from sympy import totient, divisors
def A026119(n):
    m = (n<<1)+1
    return sum(totient(d)<Chai Wah Wu, Feb 21 2023

a(n) = (Sum_{d | 2n+1} phi(d)*2^((2n+1)/d)) / (4n+2).

A066313 Number of aperiodic necklaces with n red or blue beads such that two necklaces are equivalent under the operation of simultaneously turning the necklace over and switching the two colors.

Original entry on oeis.org

1, 1, 1, 2, 3, 6, 9, 18, 28, 57, 93, 181, 315, 612, 1091, 2100, 3855, 7392, 13797, 26436, 49929, 95790, 182361, 350440, 671088, 1292445, 2485504, 4797261, 9256395, 17903316, 34636833, 67124160, 130150493, 252675975, 490853403, 954498874, 1857283155, 3616938738

Offset: 1

Christian G. Bower, Dec 13 2001; revised Apr 25 2006

nonn

Also number of aperiodic cyclic graphs with oriented edges on n nodes that can be turned over.

			The equivalence requires the "turning over" operation and the "switching colors" operation to be simultaneous; thus rrrbbrrb is equivalent to rbbrrbbb, but not to bbbrrbbr.

Andrew Howroyd, Table of n, a(n) for n = 1..1000
N. J. A. Sloane, Transforms
Index entries for sequences related to bracelets

Cf. A053656.

\\ here b(n) is A053656.
b(n)={(sumdiv(n, d, eulerphi(d)*2^(n/d))/n + if(n%2==0, 2^(n/2-1)))/2}
a(n)={sumdiv(n, d, moebius(d)*b(n/d))} \\ Andrew Howroyd, Jun 14 2021

Moebius transform of A053656.

Terms a(36) and beyond from Andrew Howroyd, Jun 14 2021

cp's OEIS Frontend

A256216 a(n) = A053656(n) - A000011(n).

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A000011 Number of n-bead necklaces (turning over is allowed) where complements are equivalent.

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A123045 Number of frieze patterns of length n under a certain group (see Pisanski et al. for precise definition).

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A256217 a(n) = A000011(n) - A256216(n).

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A283846 Number of n-gonal inositol homologs with 2 kinds of achiral proligands.

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A283847 Number of n-gonal inositol homologs with 2 kinds of achiral proligands.

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A283848 Number of n-gonal inositol homologs with 2 kinds of achiral proligands.

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