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 A056411 Number of step cyclic shifted sequences using a maximum of three different symbols. 9
 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 (list; graph; refs; listen; history; text; internal format)
 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), ", ")); \\ 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 EXTENSIONS Added more terms, Joerg Arndt, Aug 27 2014 STATUS approved

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Last modified February 7 02:40 EST 2023. Contains 360111 sequences. (Running on oeis4.)