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

Offset: 1

Author

Marks R. Nester

Keywords

• nonn
• easy

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

Examples

			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.
		

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).

Crossrefs

Column 6 of A284855.

Cf. A029744, A038754, A056452.

Programs

Magma

[IsEven(n) select 6^(n div 2)*7/2 else 6^((n+1) div 2): n in [1..30]]; // Vincenzo Librandi, Sep 22 2018

Mathematica

LinearRecurrence[{0,6},{6,21},30] (* Harvey P. Dale, Feb 02 2015 *)
k = 6; Table[(k^Floor[(n + 1)/2] + k^Ceiling[(n + 1)/2]) / 2, {n, 30}] (* Robert A. Russell, Sep 21 2018 *)
If[EvenQ[#], 6^(# / 2) 7/2, 6^((# + 1) / 2)]&/@Range[30] (* Vincenzo Librandi, Sep 22 2018 *)

PARI

a(n) = if(n%2, 6^((n+1)/2), 7*6^(n/2)/2); \\ Altug Alkan, Sep 21 2018

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

Extensions

More terms from Vincenzo Librandi, Sep 22 2018