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 A056488 Number of periodic palindromes using a maximum of six different symbols. 5
 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 (list; graph; refs; listen; history; text; internal format)
 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 EXTENSIONS More terms from Vincenzo Librandi, Sep 22 2018 STATUS approved

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Last modified July 21 19:25 EDT 2019. Contains 325199 sequences. (Running on oeis4.)