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A056414
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Number of step cyclic shifted sequences using a maximum of six different symbols.
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6
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6, 21, 56, 231, 462, 4291, 6966, 57561, 188866, 1519035, 3302922, 45921281, 83747286, 933081411, 3920355712, 22075451286, 62230996506, 940379310731, 1781757016326, 22856965214727, 87052415641136, 598280600648031, 1560731765058606, 24680195365765751, 56860576713326910, 546736312124316741, 2105947271634851386
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OFFSET
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1,1
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COMMENTS
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See A056371 for an explanation of step shifts. Under step cyclic shifts, abcde, bdace, bcdea, cdeab and daceb etc. are equivalent.
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REFERENCES
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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]
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LINKS
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FORMULA
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Refer to Titsworth or slight "simplification" in Nester.
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MATHEMATICA
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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 *)
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PROG
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(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, 6), ", "));
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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