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A056486 a(n) = (9*2^n + (-2)^n)/4 for n>0.

%I A056486 #49 Mar 23 2024 18:57:48
%S A056486 4,10,16,40,64,160,256,640,1024,2560,4096,10240,16384,40960,65536,
%T A056486 163840,262144,655360,1048576,2621440,4194304,10485760,16777216,
%U A056486 41943040,67108864,167772160,268435456,671088640,1073741824,2684354560,4294967296,10737418240
%N A056486 a(n) = (9*2^n + (-2)^n)/4 for n>0.
%C A056486 Old name was: "Number of periodic palindromes using a maximum of four different symbols".
%C A056486 Number of necklaces with n beads and 4 colors that are the same when turned over and hence have reflection symmetry. - _Herbert Kociemba_, Nov 24 2016
%D A056486 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 A056486 Vincenzo Librandi, <a href="/A056486/b056486.txt">Table of n, a(n) for n = 1..1000</a>
%H A056486 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (0,4).
%F A056486 a(n) = 4^((n+1)/2) for n odd, a(n) = 4^(n/2)*5/2 for n even.
%F A056486 From _Colin Barker_, Jul 08 2012: (Start)
%F A056486 a(n) = 4*a(n-2).
%F A056486 G.f.: 2*x*(2+5*x)/((1-2*x)*(1+2*x)). (End)
%F A056486 G.f.: -1 + (1+4*x+6*x^2)/(1-4*x^2). - _Herbert Kociemba_, Nov 24 2016
%F A056486 E.g.f.: 5*sinh(x)^2 + 2*sinh(2*x). - _Ilya Gutkovskiy_, Nov 24 2016
%F A056486 a(n) = ( 4^floor((n+1)/2) + 4^ceiling((n+1)/2) )/2. - _Robert A. Russell_, Sep 21 2018
%e A056486 G.f. = 4*x + 10*x^2 + 16*x^3 + 40*x^4 + 64*x^5 + 160*x^6 + 256*x^7 + 640*x^8 + ...
%e A056486 For example, aaabbb is not a (finite) palindrome but it is a periodic palindrome.
%p A056486 A056486:=n->(9*2^n + (-2)^n)/4: seq(A056486(n), n=1..50); # _Wesley Ivan Hurt_, Nov 24 2016
%t A056486 CoefficientList[Series[-1+(1+4*x+6*x^2)/(1-4*x^2),{x,0,30}],x] (* _Herbert Kociemba_, Nov 24 2016 *)
%t A056486 k=4; Table[(k^Floor[(n+1)/2] + k^Ceiling[(n+1)/2]) / 2, {n, 1, 30}] (* _Robert A. Russell_, Sep 21 2018 *)
%o A056486 (Magma) [(9*2^n + (-2)^n)/4 : n in [1..50]]; // _Wesley Ivan Hurt_, Nov 24 2016
%o A056486 (PARI) a(n) = (9*2^n+(-2)^n)/4; \\ _Altug Alkan_, Sep 21 2018
%o A056486 (SageMath) [2^(n-2)*(9+(-1)^n) for n in range(1,51)] # _G. C. Greubel_, Mar 23 2024
%Y A056486 Cf. A029744, A038754, A056450.
%Y A056486 Column 4 of A284855.
%K A056486 nonn,easy
%O A056486 1,1
%A A056486 _Marks R. Nester_
%E A056486 Better name from _Ralf Stephan_, Jul 18 2013