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A056488

Number of periodic palindromes using a maximum of six different symbols.

6, 21, 36, 126, 216, 756, 1296, 4536, 7776, 27216, 46656, 163296, 279936, 979776, 1679616, 5878656, 10077696, 35271936, 60466176, 211631616, 362797056, 1269789696, 2176782336, 7618738176, 13060694016, 45712429056, 78364164096, 274274574336, 470184984576

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OFFSET

1,1

COMMENTS

Also number of necklaces with n beads and 6 colors that are the same when turned over and hence have reflection symmetry. - Herbert Kociemba, Nov 24 2016

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

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (0,6).

FORMULA

a(n) = 6^((n+1)/2) for n odd, a(n) = 6^(n/2)*7/2 for n even.

From Colin Barker, Jul 08 2012: (Start)

a(n) = 6*a(n-2).

G.f.: 3*x*(2+7*x)/(1-6*x^2). (End)

a(n) = (k^floor((n+1)/2) + k^ceiling((n+1)/2)) / 2, where k = 6 is the number of possible colors. - Robert A. Russell, Sep 22 2018

EXAMPLE

G.f. = 6*x + 21*x^2 + 36*x^3 + 126*x^4 + 216*x^5 + 756*x^6 + 1296*x^7 + ...

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