

A056493


Number of primitive (period n) periodic palindromes using a maximum of two different symbols.


3



2, 1, 2, 3, 6, 7, 14, 18, 28, 39, 62, 81, 126, 175, 246, 360, 510, 728, 1022, 1485, 2030, 3007, 4094, 6030, 8184, 12159, 16352, 24381, 32766, 48849, 65534, 97920, 131006, 196095, 262122, 392364, 524286, 785407, 1048446, 1571310, 2097150, 3143497
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,1


COMMENTS

For example, aaabbb is not a (finite) palindrome but it is a periodic palindrome.
Also number of aperiodic necklaces (Lyndon words) with two colors that are the same when turned over.


REFERENCES

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia.


LINKS

Table of n, a(n) for n=1..42.
Index entries for sequences related to Lyndon words


FORMULA

Sum mu(d)*A029744(n/d) where d divides n.
From Herbert Kociemba, Nov 29 2016: (Start)
More generally, gf(k) is the g.f. for the number of necklaces with reflectional symmetry but no rotational symmetry and beads of k colors.
gf(k): Sum_{n>=1} mu(n)*Sum_{i=0..2} binomial(k,i)x^(n*i)/(1k*x^(2*n)). (End)


EXAMPLE

a(1) = 2 with aaa... and bbb..., a(2) = 1 with ababab..., a(3) = 2 with aabaab... and abbabb...., a(4) = 3 with aaabaaab... and aabbaabb... and abbbabbb....  Michael Somos, Nov 29 2016


MATHEMATICA

mx=40; gf[x_, k_]:=Sum[ MoebiusMu[n]*Sum[Binomial[k, i]x^(n i), {i, 0, 2}]/( 1k x^(2n)), {n, mx}]; CoefficientList[Series[gf[x, 2], {x, 0, mx}], x] (* Herbert Kociemba, Nov 29 2016 *)


CROSSREFS

Cf. A056458.
Sequence in context: A108618 A097719 A249050 * A277619 A001371 A277629
Adjacent sequences: A056490 A056491 A056492 * A056494 A056495 A056496


KEYWORD

nonn,changed


AUTHOR

Marks R. Nester (nesterm(AT)dpi.qld.gov.au)


EXTENSIONS

More terms and additional comments from Christian G. Bower, Jun 22 2000


STATUS

approved



