This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A056487 #67 Jun 07 2023 08:31:30
%S A056487 1,3,5,15,25,75,125,375,625,1875,3125,9375,15625,46875,78125,234375,
%T A056487 390625,1171875,1953125,5859375,9765625,29296875,48828125,146484375,
%U A056487 244140625,732421875,1220703125,3662109375,6103515625,18310546875,30517578125,91552734375
%N A056487 a(n) = 5^(n/2) for n even, a(n) = 3*5^((n-1)/2) for n odd.
%C A056487 Apparently identical to A111386! Is this a theorem? - _Klaus Brockhaus_, Jul 21 2009
%C A056487 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 A056487 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 A056487 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (0,5).
%F A056487 a(n+2) = 5*a(n), a(0)=1, a(2)=3.
%F A056487 Binomial transform of A087205. Binomial transform is A087206. - _Paul Barry_, Aug 25 2003
%F A056487 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 A056487 2nd inverse binomial transform of Fibonacci(3n+2). - _Paul Barry_, Apr 16 2004
%F A056487 a(n+3) = a(n+2)*a(n+1)/a(n). - _Reinhard Zumkeller_, Mar 04 2011
%F A056487 a(n) = 3^((1 - (-1)^n)/2) * 5^((2*n + (-1)^n-1)/4). - _Bruno Berselli_, Mar 24 2011
%F A056487 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 A056487 E.g.f.: cosh(sqrt(5)*x) + 3*sinh(sqrt(5)*x)/sqrt(5). - _Stefano Spezia_, Jun 06 2023
%p A056487 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 A056487 Table[3^((1 - (-1)^n)/2)*5^((2*n + (-1)^n - 1)/4), {n, 0, 30}] (* _Wesley Ivan Hurt_, Nov 24 2016 *)
%t A056487 CoefficientList[Series[(1 + 3 x)/(1 - 5 x^2), {x, 0, 31}], x] (* _Michael De Vlieger_, Nov 24 2016 *)
%t A056487 LinearRecurrence[{0, 5}, {1, 3}, 35] (* _Vincenzo Librandi_, Nov 25 2016 *)
%t A056487 k=5; Table[(k^Floor[(n+1)/2] + k^Ceiling[(n+1)/2]) / 2, {n, -1, 30}] (* _Robert A. Russell_, Sep 21 2018 *)
%o A056487 (Magma) [n le 2 select 2*n-1 else 5*Self(n-2): n in [1..28]]; // _Bruno Berselli_, Mar 24 2011
%o A056487 (PARI) a(n)=if(n%2,3,1)*5^(n\2) \\ _Charles R Greathouse IV_, Oct 07 2015
%o A056487 (Python)
%o A056487 def A056487(n): return 5**(n>>1)*(3 if n&1 else 1) # _Chai Wah Wu_, Oct 27 2022
%Y A056487 Column 5 of A284855.
%Y A056487 Cf. A029744, A038754, A056391, A056451, A087205, A087206, A111386.
%K A056487 nonn,easy
%O A056487 0,2
%A A056487 _Marks R. Nester_
%E A056487 Changed one 'even' to 'odd' in the definition. - _R. J. Mathar_, Oct 06 2010