A032279 -id:A032279 - cp's OEIS frontend

A005516 Number of n-bead bracelets (turnover necklaces) with 12 red beads.

1, 1, 7, 19, 72, 196, 561, 1368, 3260, 7105, 14938, 29624, 56822, 104468, 186616, 322786, 544802, 896259, 1444147, 2278640, 3532144, 5380034, 8070400, 11926928, 17393969, 25042836, 35638596, 50152013, 69855536

Offset: 12

N. J. A. Sloane

nonn

From Vladimir Shevelev, Apr 23 2011: (Start)
Also number of non-equivalent (turnover) necklaces of 12 beads each of them painted by one of n colors.
The sequence solves the so-called Reis problem about convex k-gons in case k=12 (see our comment to A032279). (End)

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

Andrew Howroyd, Table of n, a(n) for n = 12..1000
H. Gupta, Enumeration of incongruent cyclic k-gons, Indian J. Pure and Appl. Math., 10 (1979), no. 8, 964-999.
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)
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]
V. Shevelev, Necklaces and convex k-gons, Indian J. Pure and Appl. Math., 35 (2004), no. 5, 629-638.
V. Shevelev, Spectrum of permanent's values and its extremal magnitudes in Lambda_n^3 and Lambda_n(alpha,beta,gamma), arXiv:1104.4051 [math.CO], 2011. (Cf. Section 5.)
A. P. Street, Letter to N. J. A. Sloane, N.D.
Index entries for sequences related to bracelets
Index entries for linear recurrences with constant coefficients, signature (4,-4,-2,4,-4,12,-12,2,2,-12,24,-18,4,4,-6,15,-20,0,10,-4,10,0,-20,15,-6,4,4,-18,24,-12,2,2,-12,12,-4,4,-2,-4,4,-1).

Column k=12 of A052307.

k = 12; Table[(Apply[Plus, Map[EulerPhi[ # ]Binomial[n/#, k/# ] &, Divisors[GCD[n, k]]]]/n + Binomial[If[OddQ[n], n - 1, n - If[OddQ[k], 2, 0]]/2, If[OddQ[k], k - 1, k]/2])/2, {n, k, 50}] (* Robert A. Russell, Sep 27 2004 *)
k=12;CoefficientList[Series[x^k*(1/k Plus@@(EulerPhi[#] (1-x^#)^(-(k/#))&/@Divisors[k])+(1+x)/(1-x^2)^Floor[k/2+1])/2,{x,0,50}],x] (* Herbert Kociemba, Nov 04 2016 *)

Let s(n,k,d) = 1, if n==k (mod d), s(n,k,d) = 0, otherwise. Then a(n) = s(n,0,12)/6 + (n-6)*s(n,0,6)/72 + (n-4)*(n-8)*s(n,0,4)/384 + (n-3)*(n-6)*(n-9)*s(n,0,3)/1944 + (3840*C(n-1,11) + (n+1)*(n-2)*(n-4)*(n-6)*(n-8)*(n-10))/92160, if n is even; a(n) = (n-3)*(n-6)*(n-9)*s(n,0,3)/1944 + (3840*C(n-1,11) + (n-1)*(n-3)*(n-5)*(n-7)*(n-9)*(n-11))/92160, if n is odd. - Vladimir Shevelev, Apr 23 2011
From Herbert Kociemba, Nov 04 2016: (Start)
G.f.: 1/2*x^12*((1+x)/(1-x^2)^7 + 1/12*(1/(-1+x)^12 + 1/(-1+x^2)^6 + 2/(-1+x^3)^4 - 2/(-1+x^4)^3 + 2/(-1+x^6)^2 - 4/(-1+x^12))).
G.f.: k=12, x^k*((1/k)*(Sum_{d|k} phi(d)*(1 - x^d)^(-k/d)) + (1 + x)/(1 -x^2)^floor((k+2)/2))/2. (End)

Sequence extended and description corrected by Christian G. Bower

A228707 G.f.: (1-3x+5x^2-5x^3+5x^4-5x^5+5x^6-3x^7+x^8)/((1-x)^4(1+x^4)*(1+x^2)^2).

Original entry on oeis.org

1, 1, 1, 3, 6, 8, 10, 16, 24, 29, 35, 47, 61, 72, 84, 104, 127, 145, 165, 195, 228, 256, 286, 328, 374, 413, 455, 511, 571, 624, 680, 752, 829, 897, 969, 1059, 1154, 1240, 1330, 1440, 1556, 1661, 1771, 1903, 2041, 2168, 2300, 2456, 2619, 2769

Offset: 0

N. J. A. Sloane, Sep 06 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
E. Kirkman, J. Kuzmanovich and J. J. Zhang, Invariants of (-1)-Skew Polynomial Rings under Permutation Representations, arXiv preprint arXiv:1305.3973, 2013

Cf. A032279, A228706.

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-3*x+5*x^2-5*x^3+5*x^4-5*x^5+5*x^6-3*x^7+x^8)/((1-x)^4*(1+x^4)*(1+x^2)^2))); // Vincenzo Librandi, Sep 07 2013

CoefficientList[Series[(1 - 3 x + 5 x^2 - 5 x^3 + 5 x^4 - 5 x^5 + 5 x^6 - 3 x^7 + x^8) / ((1 - x)^4 (1 + x^4) (1 + x^2)^2), {x, 0, 50}],x] (* Vincenzo Librandi, Sep 07 2013 *)

G.f.: (1-x+x^2)*(1-2 *x+2*x^2-x^3+2*x^4-2*x^5+x^6)/((1+x^2)^2*(1-x)^4*(1+x^4)).

A143654 Array T(n,k) read by rows: number of binary bracelets with n beads, k of them 0, with 00 prohibited, (n >= 2, 0 <= k <= floor(n/2)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 3, 3, 1, 1, 1, 4, 4, 3, 1, 1, 1, 4, 5, 4, 1, 1, 1, 5, 7, 8, 3, 1, 1, 1, 5, 8, 10, 5, 1, 1, 1, 6, 10, 16, 10, 4, 1, 1, 1, 6, 12, 20, 16, 7, 1, 1, 1, 7, 14, 29, 26, 16, 4, 1, 1, 1, 7, 16, 35, 38, 26, 8, 1, 1, 1, 8, 19, 47, 57, 50

Offset: 2

Washington Bomfim, Aug 28 2008

The initial columns give A057427, A057427, A004526, A069905, A005232, A032279, A005513, A032280, A005514, A032281, A005515, A032282, A005516. Row sums give A129526.
A binary bracelet with n beads, k of them 0, with 00 prohibited has from 0 to floor(n/2) beads 0, i.e., 0 <= k <= floor(n/2). If n is even, the bracelet 0101...01 with n/2 beads of each kind does not have 00 and we cannot change any 1 of it to a 0. If n is odd we cannot change a 1 to a 0 in the bracelet 0101...011 with (n-1)/2 beads 0.
The number of binary bracelets with n beads, 0 <= k <= floor(n/2) of them 0 with 00 prohibited, is equal to the number of binary bracelets with n-k beads, k of them 0. See below.
Let B be a binary bracelet with n-k beads, k of them 0. If we insert one 1 (circularly) after a 0 of B, we obtain a bracelet with n-k+1 beads, k of them 0.
If we do this insertion k times, each time after a distinct 0 of B, we obtain a bracelet with n = n-k+k beads, k of them 0, with 00 prohibited.
On the contrary, Let B be a binary bracelet with n beads, k of them 0, with 00 prohibited. If we remove from B one 1 that is after a 0, we obtain a bracelet of n-1 beads, k of them 0. (If not and we undo the removal, the configuration obtained cannot be a bracelet and this is absurd.) If we repeat this removal k times, after each distinct bead 0, we obtain a bracelet with n-k beads, k of them 0.

			Array begins
1 1
1 1
1 1 1
1 1 1
1 1 2 1
1 1 2 1
1 1 3 2 1
1 1 3 3 1
1 1 4 4 3 1
...
A129526(10) = A057427(10) + A057427(9) + A004526(8) + A069905(7) + A005232(6) +
A032279(5) = 1+1+4+4+3+1 = 14.

Cf. A057427, A004526, A069905, A005232, A032279, A005513, A032280, A005514, A032281, A005515, A032282, A005516. Row sums of array give A129526.

cp's OEIS Frontend

A005516 Number of n-bead bracelets (turnover necklaces) with 12 red beads.

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A228707 G.f.: (1-3x+5x^2-5x^3+5x^4-5x^5+5x^6-3x^7+x^8)/((1-x)^4(1+x^4)*(1+x^2)^2).

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A143654 Array T(n,k) read by rows: number of binary bracelets with n beads, k of them 0, with 00 prohibited, (n >= 2, 0 <= k <= floor(n/2)).

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cp's OEIS Frontend

A005516 Number of n-bead bracelets (turnover necklaces) with 12 red beads.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A228707 G.f.: (1-3*x+5*x^2-5*x^3+5*x^4-5*x^5+5*x^6-3*x^7+x^8)/((1-x)^4*(1+x^4)*(1+x^2)^2).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

Formula

A143654 Array T(n,k) read by rows: number of binary bracelets with n beads, k of them 0, with 00 prohibited, (n >= 2, 0 <= k <= floor(n/2)).

Views

Author

Keywords

Comments

Examples

Crossrefs

A228707 G.f.: (1-3x+5x^2-5x^3+5x^4-5x^5+5x^6-3x^7+x^8)/((1-x)^4(1+x^4)*(1+x^2)^2).