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

1,1

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.

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 *)

Prog

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

Crossrefs

Column 6 of A284855.

Cf. A029744, A038754, A056452.

Sequence in context: A151943 A207339 A284988 * A031042 A064431 A031094

Adjacent sequences: A056485 A056486 A056487 * A056489 A056490 A056491

Keyword

nonn,easy

Author

Marks R. Nester

Extensions

More terms from Vincenzo Librandi, Sep 22 2018

Status

approved