

A056371


Number of step shifted (decimated) sequences using a maximum of two different symbols.


87



2, 4, 6, 12, 12, 40, 28, 96, 104, 280, 216, 1248, 704, 2800, 4344, 8928, 8232, 44224, 29204, 136032, 176752, 419872, 381492, 2150400, 1678256, 5594000, 7461168, 22553408, 19175160, 134391040, 71585136, 269510016, 429726240, 1073758360
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OFFSET

1,1


COMMENTS

All step shifts of a sequence are considered to be equivalent, where a step shift transformation is obtained by selecting every kth element of a sequence for some k relatively prime to n. For example, 2 is relatively prime to 5 and a 2step shift of abcde is bdace.


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

G. C. Greubel, Table of n, a(n) for n = 1..1000
R. C. Titsworth, Equivalence classes of periodic sequences, Illinois J. Math., 8 (1964), 266270.


FORMULA

The cycle index is implicit in Titsworth.
a(n) = ( Sum_{k=1..n : gcd(k,n)=1} 2^( Sum_{dn} A000010(d)/ord_d(k) ) ) / A000010(n), where ord_d(k) is the multiplicative order of k modulo d.  Max Alekseyev, Jun 18 2007, corrected Nov 08 2007


MATHEMATICA

a[m_, n_] := (1/EulerPhi[n])*Sum[If[GCD[k, n] == 1, m^DivisorSum[n, EulerPhi[#] / MultiplicativeOrder[k, #]&], 0], {k, n}]; Table[a[2, n], {n, 34}] (* JeanFrançois Alcover, Dec 04 2015 *)


PROG

(PARI) { a(n) = sum(k=1, n, if(gcd(k, n)==1, 2^sumdiv(n, d, eulerphi(d)/znorder(Mod(k, d))), 0); ) / eulerphi(n) } /* Max Alekseyev, Jun 18 2007 */


CROSSREFS

Cf. A002729.
A row or column of A132191.
Sequence in context: A137387 A137394 A062856 * A271822 A067874 A015733
Adjacent sequences: A056368 A056369 A056370 * A056372 A056373 A056374


KEYWORD

nonn


AUTHOR

Marks R. Nester


EXTENSIONS

More terms from Max Alekseyev, Jun 18 2007


STATUS

approved



