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%I A056414 #27 Aug 22 2017 20:53:11
%S A056414 6,21,56,231,462,4291,6966,57561,188866,1519035,3302922,45921281,
%T A056414 83747286,933081411,3920355712,22075451286,62230996506,940379310731,
%U A056414 1781757016326,22856965214727,87052415641136,598280600648031,1560731765058606,24680195365765751,56860576713326910,546736312124316741,2105947271634851386
%N A056414 Number of step cyclic shifted sequences using a maximum of six different symbols.
%C A056414 See A056371 for an explanation of step shifts. Under step cyclic shifts, abcde, bdace, bcdea, cdeab and daceb etc. are equivalent.
%D A056414 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]
%H A056414 D. Z. Dokovic, I. Kotsireas et al., <a href="http://arxiv.org/abs/1405.7328">Charm bracelets and their application to the construction of periodic Golay pairs</a>, arXiv:1405.7328 [math.CO], 2014.
%H A056414 R. C. Titsworth, <a href="http://projecteuclid.org/euclid.ijm/1256059671">Equivalence classes of periodic sequences</a>, Illinois J. Math., 8 (1964), 266-270.
%F A056414 Refer to Titsworth or slight "simplification" in Nester.
%t A056414 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, 6]]; cb, {n, 1, 27}] (* _Jean-François Alcover_, Dec 04 2015, after _Joerg Arndt_ *)
%o A056414 (PARI)  \\ see p.3 of the Dokovic et al. reference
%o A056414 M(j,  L)={my(m=1); while ( sum(i=0, m-1, j^i) % L != 0, m+=1 ); m; }
%o A056414 c(j, t, n)=sum(u=0,n-1, 1/M(j, n / gcd(n, u*(j-1)+t) ) );
%o A056414 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) ) ) );
%o A056414 for(n=1, 66, print1(CB(n,6),", "));
%o A056414 \\ _Joerg Arndt_, Aug 27 2014
%Y A056414 Row 6 of A285548.
%Y A056414 Cf. A002729.
%K A056414 nonn
%O A056414 1,1
%A A056414 _Marks R. Nester_
%E A056414 Added more terms, _Joerg Arndt_, Aug 27 2014

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