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A091696 Number of classes of compositions of n equivalent under reflection or cycling. 5
1, 2, 3, 5, 7, 12, 17, 29, 45, 77, 125, 223, 379, 686, 1223, 2249, 4111, 7684, 14309, 27011, 50963, 96908, 184409, 352697, 675187, 1296857, 2493725, 4806077, 9272779, 17920859, 34669601, 67159049, 130216123, 252745367, 490984487, 954637557, 1857545299 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..1000

FORMULA

a(n) = A000029(n) - 1.

a(n) = A056342(n) + 1.

G.f.: ( Sum_{n>=1} phi(n)*log(2+1/(-1+x^n))/n + (1-1x^2+x^3)/((x-1)*(1-2*x^2)) )/(-2). - Herbert Kociemba, Dec 04 2016

EXAMPLE

7 has 15 partitions and 64 compositions. Compositions can be mapped to other compositions by reflection, cycling, or both, e.g., {1,2,4} -> {4,2,1} (reflection), {2,4,1} (cycling), or {1,4,2} (both); this defines the equivalence relation used. The number of equivalence classes so defined is 2 greater than the number of partitions because only {3,1,2,1} and {2,1,2,1,1} (and their equivalents) cannot be mapped to the conventionally stated forms of partitions (here, {3,2,1,1} and {2,2,1,1,1} respectively). So a(7) = 15 + 2 = 17.

MAPLE

with(numtheory):

a:= n-> add(phi(d)*2^(n/d)/(2*n), d=divisors(n))

        +`if`(irem(n, 2)=0, 2^(n/2-1) +2^(n/2-2), 2^((n-1)/2)) -1:

seq(a(n), n=1..40);  # Alois P. Heinz, Oct 20 2012

MATHEMATICA

Needs["Combinatorica`"]

nn=40; Apply[Plus, Table[CoefficientList[Series[CycleIndex[DihedralGroup[n], s]/.Table[s[i]->x^i/(1-x^i), {i, 1, nn}], {x, 0, nn}], x], {n, 1, nn}]]  (* Geoffrey Critzer, Oct 18 2012 *)

mx:=50; CoefficientList[Series[(Sum[(EulerPhi[n] Log[2+1/(-1+x^n)])/n, {n, 1, mx}]+(1-1x^2+ x^3)/((x-1) (1-2 x^2)))/(-2), {x, 0, mx}], x] (* Herbert Kociemba, Dec 04 2016 *)

a[n_] := (1/4)*(Mod[n, 2] + 3)*2^Quotient[n, 2] + DivisorSum[n, EulerPhi[#]*2^(n/#) & ]/(2*n) - 1; Array[a, 37] (* Jean-Fran├žois Alcover, Nov 05 2017 *)

CROSSREFS

Cf. A000029, A056342.

Sequence in context: A206788 A002965 A206290 * A280303 A048808 A263358

Adjacent sequences:  A091693 A091694 A091695 * A091697 A091698 A091699

KEYWORD

nonn

AUTHOR

Neil Fernandez, Jan 29 2004

EXTENSIONS

More terms from Sean A. Irvine, Feb 09 2012

STATUS

approved

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Last modified May 21 23:35 EDT 2019. Contains 323470 sequences. (Running on oeis4.)