A045629 -id:A045629 - cp's OEIS frontend

A006840 Number of 2n-bead black-white reversible complementable necklaces with n black beads.

1, 1, 2, 3, 7, 13, 35, 85, 257, 765, 2518, 8359, 28968, 101340, 361270, 1297879, 4707969, 17179435, 63068876, 232615771, 861725794, 3204236779, 11955836258, 44748176653, 167959144032, 632058070310, 2384235077576, 9013628451275

Offset: 0

N. J. A. Sloane

nonn

J. A. Hoskins, C. E. Praeger and A. P. Street, Balanced twills with bounded float length, Congress. Numerantium, 40 (1983), 77-89.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

G. C. Greubel, Table of n, a(n) for n = 0..1650 (terms 0..200 from Andrew Howroyd)
J. A. Hoskins, C. E. Praeger and A. P. Street, Balanced twills with bounded float length, Congress. Numerantium, 40 (1983), 77-89. (Annotated scanned copy)
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)
A. P. Street, Letter to N. J. A. Sloane, N.D.
Index entries for sequences related to bracelets

Row sums of A259341.

Cf. A045629.

b[n_] := (1/(2*n))*DivisorSum[n, EulerPhi[n/#]*Binomial[2*# - 1, # - 1] + EulerPhi[2*(n/#)]*2^(# - 1) &]; a[0] = 1; a[n_] := (b[n] + 2^(n-2) + Binomial[n - Mod[n, 2], Quotient[n, 2]]/2)/2; Table[a[n], {n, 0, 27}] (* Jean-François Alcover, Oct 08 2017, after Andrew Howroyd *)

\\ here b is A045629
b(n) = (1/(2*n)) * sumdiv(n, d, eulerphi(n/d)*binomial(2*d-1, d-1) + eulerphi(2*n/d)*2^(d-1));
a(n) = if(n==0, 1, (b(n) + 2^(n-2) + binomial(n-n%2, n\2)/2) / 2); \\ Andrew Howroyd, Sep 27 2017

If n is odd, a(n) = (1/2) * (A045629 + (1/2) * C(n-1, (n-1)/2) + 2^(n-2)); if n is even, a(n) = (1/2) * (A045629 + (1/2) * C(n, n/2) + 2^(n-2)). - Christian G. Bower

More terms from David W. Wilson

A123717 Signature permutation of a Catalan automorphism: row 253 of table A122203.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 5, 8, 7, 9, 11, 14, 16, 19, 10, 15, 13, 21, 22, 12, 20, 17, 18, 23, 25, 28, 30, 33, 37, 39, 42, 44, 47, 51, 53, 56, 60, 24, 29, 38, 43, 52, 27, 41, 35, 58, 59, 36, 62, 63, 64, 26, 40, 34, 57, 61, 31, 54, 45, 46, 32, 55, 48, 49, 50, 65, 67, 70, 72, 75, 79, 81

Offset: 0

Antti Karttunen, Oct 11 2006

nonn

This is the signature-permutation of Catalan automorphism which is derived from nonrecursive Catalan automorphism *A123503 with the recursion schema SPINE (defined in A122203).

The number of fixed points in range [A014137(n-1)..A014138(n-1)] of this permutation begins as 1,1,2,1,3,1,4,1,8,1,16,1,47,..., the LCM of cycle sizes as 1,1,1,2,12,12,120,120,840,840,5040,5040,55440,... (cf. A089423) and the cycle-count sequence seems to be A045629. (To be proved.)

A. Karttunen, paper in preparation, draft available by e-mail.

Index entries for signature-permutations of Catalan automorphisms

Inverse: A123718. a(n) = A057509(A089854(n)). Row 253 of A122203.

A045632 Number of 2n-bead black-white complementable necklaces with n black beads and fundamental period 2n.

Original entry on oeis.org

1, 1, 1, 2, 5, 14, 40, 127, 408, 1364, 4638, 16079, 56401, 200169, 716599, 2585833, 9392704, 34319666, 126046498, 465076159, 1723092407, 6407856762, 23910255143, 89493903437, 335912684824, 1264106399919, 4768447977465, 18027218146451

Offset: 0

David W. Wilson

nonn

Index entries for sequences related to Lyndon words

A045629[n_] := If[n == 0, 1, (1/(2n)) DivisorSum[n, EulerPhi[n/#] Binomial[ 2#-1, #-1] + EulerPhi[2(n/#)] 2^(#-1)&]];
a[n_] := If[n == 0, 1, Sum[MoebiusMu[n/d] A045629[d], {d, Divisors[n]}]];
Table[a[n], {n, 0, 27}] (* Jean-François Alcover, Aug 28 2019 *)

Moebius transform of A045629 (Christian Bower).

A115122 Number of imprimitive (periodic) 2n-bead black-white complementable necklaces with n black beads.

Original entry on oeis.org

0, 0, 1, 1, 2, 1, 4, 1, 7, 3, 16, 1, 49, 1, 129, 17, 415, 1, 1408, 1, 4659, 130, 16081, 1, 56858, 15, 200171, 1367, 716733, 1, 2590529, 1, 9393119, 16082, 34319668, 142, 126104312, 1, 465076161, 200172, 1723097474, 1, 6408573532, 1, 23910271229, 2587214, 89493903439

Offset: 0

Valery A. Liskovets, Jan 17 2006

a(p)=1 for prime p.

A045629[n_] := If[n==0, 1, (1/(2n)) DivisorSum[n, EulerPhi[n/#] Binomial[ 2 # - 1, # - 1] + EulerPhi[2 (n/#)] 2^(# - 1)&]];
A045632[n_] := If[n==0, 1, Sum[MoebiusMu[n/d] A045629[d], {d, Divisors[n]}]];
a[n_] := A045629[n] - A045632[n];
Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Aug 28 2019 *)

a(n)=A045629(n) - A045632(n).

More terms from Jean-François Alcover, Aug 28 2019

cp's OEIS Frontend

A006840 Number of 2n-bead black-white reversible complementable necklaces with n black beads.

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A123717 Signature permutation of a Catalan automorphism: row 253 of table A122203.

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A045632 Number of 2n-bead black-white complementable necklaces with n black beads and fundamental period 2n.

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A115122 Number of imprimitive (periodic) 2n-bead black-white complementable necklaces with n black beads.

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