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A056488 Number of periodic palindromes using a maximum of six different symbols. 5

%I

%S 6,21,36,126,216,756,1296,4536,7776,27216,46656,163296,279936,979776,

%T 1679616,5878656,10077696,35271936,60466176,211631616,362797056,

%U 1269789696,2176782336,7618738176,13060694016,45712429056,78364164096,274274574336,470184984576

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

%C 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

%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 Vincenzo Librandi, <a href="/A056488/b056488.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (0,6).

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

%F From _Colin Barker_, Jul 08 2012: (Start)

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

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

%F 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

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

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

%t LinearRecurrence[{0,6},{6,21},30] (* _Harvey P. Dale_, Feb 02 2015 *)

%t k = 6; Table[(k^Floor[(n + 1)/2] + k^Ceiling[(n + 1)/2]) / 2, {n, 30}] (* _Robert A. Russell_, Sep 21 2018 *)

%t If[EvenQ[#], 6^(# / 2) 7/2, 6^((# + 1) / 2)]&/@Range[30] (* _Vincenzo Librandi_, Sep 22 2018 *)

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

%o (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

%Y Column 6 of A284855.

%Y Cf. A029744, A038754, A056452.

%K nonn,easy

%O 1,1

%A _Marks R. Nester_

%E More terms from _Vincenzo Librandi_, Sep 22 2018

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Last modified August 26 00:35 EDT 2019. Contains 326324 sequences. (Running on oeis4.)