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A056412
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Number of step cyclic shifted sequences using a maximum of four different symbols.
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8
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4, 10, 20, 55, 76, 430, 460, 2605, 5164, 26962, 38572, 367645, 431780, 3203430, 8993804, 33860125, 63177820, 636462350, 803796700, 6886280971, 17456594380, 79965550558, 139069427020, 1466861706095, 2251803181492, 14434628481170, 37066691779180, 214483458079665, 354963555781060, 4803855154772166
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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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Table of n, a(n) for n=1..30.
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.
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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, 4]]; cb, {n, 1, 30}] (* 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, 4), ", "));
\\ Joerg Arndt, Aug 27 2014
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CROSSREFS
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Row 4 of A285548.
Cf. A002729.
Sequence in context: A019498 A237626 A020149 * A032275 A220828 A015220
Adjacent sequences: A056409 A056410 A056411 * A056413 A056414 A056415
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KEYWORD
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nonn
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AUTHOR
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Marks R. Nester
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EXTENSIONS
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Added more terms, Joerg Arndt, Aug 27 2014
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STATUS
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approved
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