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a(n) = 5^(n/2) for n even, a(n) = 3*5^((n-1)/2) for n odd.
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%I #67 Jun 07 2023 08:31:30

%S 1,3,5,15,25,75,125,375,625,1875,3125,9375,15625,46875,78125,234375,

%T 390625,1171875,1953125,5859375,9765625,29296875,48828125,146484375,

%U 244140625,732421875,1220703125,3662109375,6103515625,18310546875,30517578125,91552734375

%N a(n) = 5^(n/2) for n even, a(n) = 3*5^((n-1)/2) for n odd.

%C Apparently identical to A111386! Is this a theorem? - _Klaus Brockhaus_, Jul 21 2009

%C For n > 1, number of necklaces with n-1 beads and 5 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 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (0,5).

%F a(n+2) = 5*a(n), a(0)=1, a(2)=3.

%F Binomial transform of A087205. Binomial transform is A087206. - _Paul Barry_, Aug 25 2003

%F G.f.: (1+3*x)/(1-5*x^2); a(n) = 5^(n/2)(1/2 + 3*sqrt(5)/10 + (1/2 - 3*sqrt(5)/10)(-1)^n). - _Paul Barry_, Mar 19 2004

%F 2nd inverse binomial transform of Fibonacci(3n+2). - _Paul Barry_, Apr 16 2004

%F a(n+3) = a(n+2)*a(n+1)/a(n). - _Reinhard Zumkeller_, Mar 04 2011

%F a(n) = 3^((1 - (-1)^n)/2) * 5^((2*n + (-1)^n-1)/4). - _Bruno Berselli_, Mar 24 2011

%F a(n+1) = (k^floor((n+1)/2) + k^ceiling((n+1)/2)) / 2, where k=5 is the number of possible colors. - _Robert A. Russell_, Sep 22 2018

%F E.g.f.: cosh(sqrt(5)*x) + 3*sinh(sqrt(5)*x)/sqrt(5). - _Stefano Spezia_, Jun 06 2023

%p A056487:=n->3^((1-(-1)^n)/2)*5^((2*n+(-1)^n-1)/4): seq(A056487(n), n=0..40); # _Wesley Ivan Hurt_, Nov 24 2016

%t Table[3^((1 - (-1)^n)/2)*5^((2*n + (-1)^n - 1)/4), {n, 0, 30}] (* _Wesley Ivan Hurt_, Nov 24 2016 *)

%t CoefficientList[Series[(1 + 3 x)/(1 - 5 x^2), {x, 0, 31}], x] (* _Michael De Vlieger_, Nov 24 2016 *)

%t LinearRecurrence[{0, 5}, {1, 3}, 35] (* _Vincenzo Librandi_, Nov 25 2016 *)

%t k=5; Table[(k^Floor[(n+1)/2] + k^Ceiling[(n+1)/2]) / 2, {n, -1, 30}] (* _Robert A. Russell_, Sep 21 2018 *)

%o (Magma) [n le 2 select 2*n-1 else 5*Self(n-2): n in [1..28]]; // _Bruno Berselli_, Mar 24 2011

%o (PARI) a(n)=if(n%2,3,1)*5^(n\2) \\ _Charles R Greathouse IV_, Oct 07 2015

%o (Python)

%o def A056487(n): return 5**(n>>1)*(3 if n&1 else 1) # _Chai Wah Wu_, Oct 27 2022

%Y Column 5 of A284855.

%Y Cf. A029744, A038754, A056391, A056451, A087205, A087206, A111386.

%K nonn,easy

%O 0,2

%A _Marks R. Nester_

%E Changed one 'even' to 'odd' in the definition. - _R. J. Mathar_, Oct 06 2010