A056413 - OEIS

A056413

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

5, 15, 35, 120, 201, 1505, 2015, 14070, 37085, 246753, 445515, 5205790, 7832185, 72703645, 254689657, 1196213445, 2805046965, 35322811755, 55770979195, 596439735024, 1892294578755, 10837223014665, 23559159229935, 310484619147940, 596046508875701, 4776013513099405, 15330413466776835, 110874578286500410

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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..28.

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, 5]]; cb, {n, 1, 28}] (* 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, 5), ", "));

\\ Joerg Arndt, Aug 27 2014

CROSSREFS

Row 5 of A285548.

Cf. A002729.

Sequence in context: A292912 A346755 A091875 * A032276 A346890 A333932

Adjacent sequences: A056410 A056411 A056412 * A056414 A056415 A056416