A256216 -id:A256216 - cp's OEIS frontend

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

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

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

A053656 Number of cyclic graphs with oriented edges on n nodes (up to symmetry of dihedral group).

Original entry on oeis.org

1, 2, 2, 4, 4, 9, 10, 22, 30, 62, 94, 192, 316, 623, 1096, 2122, 3856, 7429, 13798, 26500, 49940, 95885, 182362, 350650, 671092, 1292762, 2485534, 4797886, 9256396, 17904476, 34636834, 67126282, 130150588, 252679832, 490853416

Offset: 1

Jeb F. Willenbring (jwillenb(AT)ucsd.edu), Feb 14 2000

Also number of bracelets (or necklaces) with n red or blue beads such that the beads switch colors when bracelet is turned over.
a(n) is also the number of frieze patterns generated by filling a 1 X n block with n copies of an asymmetric motif (where the copies are chosen from original motif or a 180-degree rotated copy) and then repeating the block by translation to produce an infinite frieze pattern. (Pisanski et al.)
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. (Boldi et al.) - Sebastiano Vigna, Jan 08 2018

			2 at n=3 because there are two such cycles. On (o -> o -> o ->) and (o -> o <- o ->).

Jeb F. Willenbring, A stability result for a Hilbert series of O_n(C) invariants.

Seiichi Manyama, Table of n, a(n) for n = 1..3334
Rémi Cocou Avohou, Joseph Ben Geloun, and Reiko Toriumi, Counting U(N)^{⊗r} ⊗ O(N)^{⊗q} invariants and tensor model observables, Eur. Phys. J. C 84, 839 (2024), see pp. 11, 27; Preprint arXiv:2404.16404 [hep-th], 2024. See pp. 18, 49.
Paolo Boldi and Sebastiano Vigna, Fibrations of Graphs, Discrete Math., 243 (2002), 21-66.
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. 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.
Jeb F. Willenbring, Home page.
A. Yajima, How to calculate the number of stereoisomers of inositol-homologs, Bull. Chem. Soc. Jpn. 2014, 87, 1260-1264. See Tables 1 and 2 (and text).
Index entries for sequences related to bracelets.

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

v:=proc(n) local k, t1; t1:=0; for k in divisors(n) do t1 := t1+phi(k)*2^(n/k); od: t1; end;
h:=n-> if n mod 2 = 0 then (n/2)*2^(n/2); else 0; fi;
A053656:=n->(v(n)+h(n))/(2*n); # N. J. A. Sloane, Nov 11 2006

a[n_] := Total[ EulerPhi[#]*2^(n/#)& /@ Divisors[n]]/(2n) + 2^(n/2-2)(1-Mod[n, 2]); Table[a[n], {n, 1, 35}] (* Jean-François Alcover, Nov 21 2011 *)

a(n)={(sumdiv(n, d, eulerphi(d)*2^(n/d))/n + if(n%2==0, 2^(n/2-1)))/2} \\ Andrew Howroyd, Jun 16 2021

G.f.: x/(1-x) + x^2/(2*(1-2*x^2)) + Sum_{n >= 1} (x^(2*n)/(2*n)) * Sum_{ d divides n } phi(d)/(1-x^d)^(2*n/d), or x^2/(2*(1-2*x^2)) - Sum_{n >= 1} phi(n)*log(1-2*x^n)/(2*n). [corrected and extended by Andrey Zabolotskiy, Oct 17 2017]

a(n) = A000031(n)/2 + (if n even) 2^(n/2-2).

More terms and additional comments from Christian G. Bower, Dec 13 2001

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.

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

A284711 Even bisection of A283848.

Original entry on oeis.org

23, 86, 339, 1332, 5298, 21066, 83987, 334966, 1336988, 5338206, 21321234, 85176636, 340338398, 1360073016, 5435820051, 21727481616, 86853790498, 347214198246, 1388133456348, 5549915835836, 22190143855898, 88725807876186, 354775752246802, 1418633882621748, 5672803378074548

Offset: 2

N. J. A. Sloane, Apr 01 2017

nonn

Robert Israel, Table of n, a(n) for n = 2..1659
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.

The even bisection of A283848 gives A000079.

f:= proc(n) uses numtheory;
(4*n)^(-1)*add(phi(d)*4^(2*n/d),d=select(type,divisors(2*n),even))+5*2^(2*n-2)
end proc:
map(f, [$2..40]);

A(m,n) = if (m%2, 2^((m-1)/2)*n^((m+1)/2), sumdiv(m, d, ((d%2)==0)*(eulerphi(d)*2^(m/d)*n^(m/d)))/(2*m) + 2^(m/2-2)*n^(m/2)*(2*n+1));
lista(nn) = for(n=2, nn, print1(T(2*n, 2), ", ")) \\ Michel Marcus, Apr 02 2017

a(n) = A283848(2*n)=(4*n)^(-1)*Sum_{d|2*n, d even} phi(d)*4^(2*n/d) + 5*2^(2*n-2). - Robert Israel, Aug 23 2018 after Fujita (2017), Eq. (101)

More terms from Michel Marcus, Apr 02 2017

Edited by Robert Israel, Aug 23 2018

cp's OEIS Frontend

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

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

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A053656 Number of cyclic graphs with oriented edges on n nodes (up to symmetry of dihedral group).

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

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Maple

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

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Maple

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

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Maple