|
| |
|
|
A056411
|
|
Number of step cyclic shifted sequences using a maximum of three different symbols.
|
|
9
|
|
|
|
3, 6, 10, 21, 24, 92, 78, 327, 443, 1632, 1698, 12769, 10464, 57840, 122822, 348222, 476052, 3597442, 3401970, 22006959, 41597374, 142677588, 186077886, 1476697627, 1694658003, 8147282460, 15690973754, 68149816689, 84520682160, 857935531804, 664166389302, 3620293575942, 8422974597554, 30656600391720, 59561470990362
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
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..35.
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, 3]]; cb, {n, 1, 35}] (* 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, 3), ", "));
\\ second argument k=3, 4, 5, 6 respectively gives A056411, A056412, A056413, A056414.
\\ Joerg Arndt, Aug 27 2014
|
|
|
CROSSREFS
|
Row 3 of A285548.
Cf. A002729.
Cf. A056412, A056413, A056414.
Sequence in context: A343386 A068865 A060179 * A068855 A068882 A076713
Adjacent sequences: A056408 A056409 A056410 * A056412 A056413 A056414
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
Marks R. Nester
|
|
|
EXTENSIONS
|
Added more terms, Joerg Arndt, Aug 27 2014
|
|
|
STATUS
|
approved
|
| |
|
|
