0,3

a(n-4)=number of necklaces with 4 black beads and n-4 white beads.

Also nonnegative integer 2 X 2 matrices with sum of elements equal to n, up to rotational symmetry.

The g.f. is Z(C_4,x), the 4-variate cycle index polynomial for the cyclic group C_4, with substitution x[i]->1/(1-x^i), i=1,...,4. Therefore by Polya enumeration a(n) is the number of cyclically inequivalent 4-necklaces whose 4 beads are labeled with nonnegative integers such that the sum of labels is n, for n=0,1,2,... See A102190 for Z(C_4,x). W. Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_de), Feb 15 2005.

D. J. Benson, Polynomial Invariants of Finite Groups, Cambridge, 1993, p. 104.

E. V. McLaughlin, Numbers of factorizations in non-unique factorial domains, Senior Thesis, Allegeny College, Meadville, PA, April 2004.

Index entries for Molien series

Index entries for sequences related to necklaces

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

There are 10 inequivalent nonnegative integer 2 X 2 matrices with sum of elements equal to 4, up to rotational symmetry:

[0 0] [0 0] [0 0] [0 0] [0 1] [0 1] [0 1] [0 2] [0 2] [1 1]

[0 4] [1 3] [2 2] [3 1] [1 2] [2 1] [3 0] [1 1] [2 0] [1 1].

1/(1-x)/(1-x^2)^2/(1-x^4)*(1+2*x^3+x^4);

k = 4; Table[Apply[Plus, Map[EulerPhi[ # ]Binomial[n/#, k/# ] &, Divisors[GCD[n, k]]]]/n, {n, k, 30}] - Robert A. Russell (russell(AT)post.harvard.edu), Sep 27 2004

Cf. A000031, A047996, A005232, A008804.

Sequence in context: A001767 A048214 A001841 this_sequence A078411 A137630 A092269

Adjacent sequences: A008607 A008608 A008609 this_sequence A008611 A008612 A008613

nonn,easy

N. J. A. Sloane (njas(AT)research.att.com).

Comment and example from Vladeta Jovovic (vladeta(AT)eunet.rs), May 18 2000