OFFSET
1,1
COMMENTS
All step shifts of a sequence are considered to be equivalent, where a step shift transformation is obtained by selecting every k-th element of a sequence for some k relatively prime to n. For example, 2 is relatively prime to 5 and a 2-step shift of abcde is bdace.
REFERENCES
M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..1000
R. C. Titsworth, Equivalence classes of periodic sequences, Illinois J. Math., 8 (1964), 266-270.
FORMULA
The cycle index is implicit in Titsworth.
a(n) = ( Sum_{k=1..n : gcd(k,n)=1} 2^( Sum_{d|n} A000010(d)/ord_d(k) ) ) / A000010(n), where ord_d(k) is the multiplicative order of k modulo d. - Max Alekseyev, Jun 18 2007, corrected Nov 08 2007
MATHEMATICA
a[m_, n_] := (1/EulerPhi[n])*Sum[If[GCD[k, n] == 1, m^DivisorSum[n, EulerPhi[#] / MultiplicativeOrder[k, #]&], 0], {k, n}]; Table[a[2, n], {n, 34}] (* Jean-François Alcover, Dec 04 2015 *)
PROG
(PARI) { a(n) = sum(k=1, n, if(gcd(k, n)==1, 2^sumdiv(n, d, eulerphi(d)/znorder(Mod(k, d))), 0); ) / eulerphi(n) } /* Max Alekseyev, Jun 18 2007 */
CROSSREFS
KEYWORD
nonn
AUTHOR
EXTENSIONS
More terms from Max Alekseyev, Jun 18 2007
STATUS
approved