A056412 Number of step cyclic shifted sequences using a maximum of four different symbols.

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

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..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.

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, 4]]; cb, {n, 1, 30}] (* 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, 4), ", "));
\\ Joerg Arndt, Aug 27 2014

Crossrefs

Row 4 of A285548.

Cf. A002729.

Sequence in context: A019498 A237626 A020149 * A032275 A220828 A015220

Adjacent sequences: A056409 A056410 A056411 * A056413 A056414 A056415

Keyword

nonn

Author

Marks R. Nester

Extensions

Added more terms, Joerg Arndt, Aug 27 2014

Status

approved