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A032122 Number of reversible strings with n beads of 5 colors.

%I A032122 #52 Sep 08 2022 08:44:51
%S A032122 1,5,15,75,325,1625,7875,39375,195625,978125,4884375,24421875,
%T A032122 122078125,610390625,3051796875,15258984375,76294140625,381470703125,
%U A032122 1907349609375,9536748046875,47683720703125
%N A032122 Number of reversible strings with n beads of 5 colors.
%H A032122 Vincenzo Librandi, <a href="/A032122/b032122.txt">Table of n, a(n) for n = 0..500</a>
%H A032122 C. G. Bower, <a href="/transforms2.html">Transforms (2)</a>
%H A032122 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (5,5,-25).
%F A032122 "BIK" (reversible, indistinct, unlabeled) transform of 5, 0, 0, 0...
%F A032122 a(n) = 1/2 * (5^n + 5^floor((n+1)/2)) = 5*A001447(n+1). - _Ralf Stephan_, Jul 07 2003
%F A032122 G.f.: (1-15*x^2) / ((1-5*x)*(1-5*x^2)). - _Colin Barker_, Jan 30 2012 [Adapted to offset 0 by _Robert A. Russell_, Nov 10 2018]
%F A032122 a(n) = 5*a(n-1) + 5*a(n-2) - 25*a(n-3). - _Vincenzo Librandi_, Jan 31 2012
%F A032122 a(n) = (A000351(n) + A056451(n)) / 2. - _Robert A. Russell_, Nov 10 2018
%e A032122 For a(2)=15, the five achiral strings are AA, BB, CC, DD, and EE; the 10 (equivalent) chiral pairs are AB-BA, AC-CA, AD-DA, AE-EA, BC-CB, BD-DB, BE-EB, CD-DC, CE-EC, and DE-ED.
%t A032122 LinearRecurrence[{5, 5, -25}, {1, 5, 15}, 31] (* _Vincenzo Librandi_, Jan 31 2012 *)
%t A032122 k=5; Table[(k^n+k^Ceiling[n/2])/2,{n,0,30}] (*_Robert A. Russell_, Nov 25 2017*)
%o A032122 (Magma) I:=[1, 5, 15]; [n le 3 select I[n] else 5*Self(n-1)+ 5*Self(n-2)-25*Self(n-3): n in [1..30]]; // _Vincenzo Librandi_, Jan 31 2012
%o A032122 (PARI) a(n)=(5^((n+1)\2)+5^n)/2 \\ _Charles R Greathouse IV_, Jan 31 2012
%Y A032122 Column 5 of A277504.
%Y A032122 Cf. A000351 (oriented), A032088(n>1) (chiral), A056451 (achiral).
%K A032122 nonn,easy
%O A032122 0,2
%A A032122 _Christian G. Bower_
%E A032122 a(0)=1 prepended by _Robert A. Russell_, Nov 10 2018