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

1, 5, 15, 75, 325, 1625, 7875, 39375, 195625, 978125, 4884375, 24421875, 122078125, 610390625, 3051796875, 15258984375, 76294140625, 381470703125, 1907349609375, 9536748046875, 47683720703125

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..500

C. G. Bower, Transforms (2)

Index entries for linear recurrences with constant coefficients, signature (5,5,-25).

Formula

"BIK" (reversible, indistinct, unlabeled) transform of 5, 0, 0, 0...

a(n) = 1/2 * (5^n + 5^floor((n+1)/2)) = 5*A001447(n+1). - Ralf Stephan, Jul 07 2003

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]

a(n) = 5*a(n-1) + 5*a(n-2) - 25*a(n-3). - Vincenzo Librandi, Jan 31 2012

a(n) = (A000351(n) + A056451(n)) / 2. - Robert A. Russell, Nov 10 2018

Example

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.

Mathematica

LinearRecurrence[{5, 5, -25}, {1, 5, 15}, 31] (* Vincenzo Librandi, Jan 31 2012 *)
k=5; Table[(k^n+k^Ceiling[n/2])/2, {n, 0, 30}] (*Robert A. Russell, Nov 25 2017*)

Prog

(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
(PARI) a(n)=(5^((n+1)\2)+5^n)/2 \\ Charles R Greathouse IV, Jan 31 2012

Crossrefs

Column 5 of A277504.

Cf. A000351 (oriented), A032088(n>1) (chiral), A056451 (achiral).

Sequence in context: A149653 A149654 A149655 * A220818 A064678 A088935

Adjacent sequences: A032119 A032120 A032121 * A032123 A032124 A032125

Keyword

nonn,easy

Author

Christian G. Bower

Extensions

a(0)=1 prepended by Robert A. Russell, Nov 10 2018

Status

approved