A056411 Number of step cyclic shifted sequences using a maximum of three different symbols.

3, 6, 10, 21, 24, 92, 78, 327, 443, 1632, 1698, 12769, 10464, 57840, 122822, 348222, 476052, 3597442, 3401970, 22006959, 41597374, 142677588, 186077886, 1476697627, 1694658003, 8147282460, 15690973754, 68149816689, 84520682160, 857935531804, 664166389302, 3620293575942, 8422974597554, 30656600391720, 59561470990362

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

Table of n, a(n) for n=1..35.

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), ", "));
\\ second argument k=3, 4, 5, 6 respectively gives A056411, A056412, A056413, A056414.
\\ Joerg Arndt, Aug 27 2014

Crossrefs

Row 3 of A285548.

Cf. A002729.

Cf. A056412, A056413, A056414.

Sequence in context: A343386 A068865 A060179 * A068855 A068882 A076713

Adjacent sequences: A056408 A056409 A056410 * A056412 A056413 A056414

Keyword

nonn

Author

Marks R. Nester

Extensions

Added more terms, Joerg Arndt, Aug 27 2014

Status

approved