OFFSET
1,1
COMMENTS
See A056371 for an explanation of step shifts. Under step cyclic shifts, abcde, bdace, bcdea, cdeab and daceb etc. are equivalent.
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
D. Z. Dokovic, I. Kotsireas et al., Charm bracelets and their application to the construction of periodic Golay pairs, arXiv:1405.7328 [math.CO], 2014.
R. C. Titsworth, Equivalence classes of periodic sequences, Illinois J. Math., 8 (1964), 266-270.
FORMULA
Refer to Titsworth or slight "simplification" in Nester.
MATHEMATICA
M[j_, L_] := Module[{m=1}, While[Sum[j^i, {i, 0, m-1}] ~Mod~ L != 0, m++]; m]; c[j_, t_, n_] := Sum[1/M[j, n/GCD[n, u*(j-1)+t]], {u, 0, n-1}]; CB[n_, k_] = If [n==1, k, 1/(n*EulerPhi[n])*Sum[If[1==GCD[n, j], k^c[j, t, n], 0], {t, 0, n-1}, {j, 1, n-1}]]; Table[Print[cb = CB[n, 3]]; cb, {n, 1, 35}] (* Jean-François Alcover, Dec 04 2015, after Joerg Arndt *)
PROG
(PARI) \\ see p.3 of the Dokovic et al. reference
M(j, L)={my(m=1); while ( sum(i=0, m-1, j^i) % L != 0, m+=1 ); m; }
c(j, t, n)=sum(u=0, n-1, 1/M(j, n / gcd(n, u*(j-1)+t) ) );
CB(n, k)=if (n==1, k, 1/(n*eulerphi(n)) * sum(t=0, n-1, sum(j=1, n-1, if(1==gcd(n, j), k^c(j, t, n), 0) ) ) );
for(n=1, 66, print1(CB(n, 3), ", "));
\\ Joerg Arndt, Aug 27 2014
CROSSREFS
KEYWORD
nonn
AUTHOR
EXTENSIONS
Added more terms, Joerg Arndt, Aug 27 2014
STATUS
approved