login
Number of primitive (period n) periodic palindromes using exactly two different symbols.
3

%I #10 Sep 29 2019 12:02:43

%S 0,1,2,3,6,7,14,18,28,39,62,81,126,175,246,360,510,728,1022,1485,2030,

%T 3007,4094,6030,8184,12159,16352,24381,32766,48849,65534,97920,131006,

%U 196095,262122,392364,524286,785407,1048446,1571310,2097150,3143497,4194302,6288381

%N Number of primitive (period n) periodic palindromes using exactly two different symbols.

%C For example, aaabbb is not a (finite) palindrome but it is a periodic palindrome.

%D 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]

%H Andrew Howroyd, <a href="/A056498/b056498.txt">Table of n, a(n) for n = 1..1000</a>

%F a(n) = Sum_{d|n} mu(d)*A027383(n/d-2) assuming that A027383(-1)=0.

%F G.f.: Sum_{k>=1} mu(k)*x^(2*k)*(1 + x^k)/((1 - x^k)*(1 - 2*x^(2*k))). - _Andrew Howroyd_, Sep 29 2019

%o (PARI) seq(n)={Vec(sum(k=1, n\2, moebius(k)*x^(2*k)*(1 + x^k)/((1 - x^k)*(1 - 2*x^(2*k))) + O(x*x^n)), -n)} \\ _Andrew Howroyd_, Sep 29 2019

%Y Column 2 of A327878.

%Y Cf. A027383, A056463.

%K nonn

%O 1,3

%A _Marks R. Nester_

%E Terms a(32) and beyond from _Andrew Howroyd_, Sep 28 2019