OFFSET
0,1
COMMENTS
a(n) is also the number of orbits of length n for T^2, if T is a map with n orbits of length n. - Thomas Ward, Apr 08 2009
REFERENCES
I. Reiten, Dynkin diagrams and the representation theory of algebras, Notices of the AMS, May 1997, Vol. 44, Number 5.
LINKS
Harvey P. Dale, Table of n, a(n) for n = 0..1000
Apisit Pakapongpun and Thomas Ward, Functorial orbit counting, Journal of Integer Sequences, 12 (2009) Article 09.2.4.
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1)
FORMULA
a(n) = 5*(2*n+1) if n even, 4*(n+1) if n odd.
G.f.: (5+8*x+5*x^2)/(1-x^2)^2.
a(n) = (1/n)*Sum_{d|n} mobius(n/d)*sigma_2(2*d). - Thomas Ward, Apr 08 2009
EXAMPLE
a(2) = (1/2)*mu(2)*sigma_2(2)+(1/2)*mu(1)*sigma_2(4) = 8. - Thomas Ward, Apr 08 2009
MATHEMATICA
CoefficientList[ Series[ (5 + 8x + 5x^2) / (1 - 2x^2 + x^4), {x, 0, 51}], x] (* Jean-François Alcover, Dec 02 2011 *)
With[{nn=40}, Riffle[10*Range[nn]-5, 8*Range[nn]]] (* or *) LinearRecurrence[ {0, 2, 0, -1}, {5, 8, 15, 16}, 80] (* Harvey P. Dale, Oct 30 2013 *)
PROG
(PARI) (1/n)*sumdiv(n, d, moebius(n/d)*sumdiv(2*d, e, e^2)) \\ Thomas Ward, Apr 08 2009
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Paul Boddington, Jan 22 2004
STATUS
approved