The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A056311

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A056311

Number of reversible strings with n beads using exactly four different colors.
(history; published version)

#26 by N. J. A. Sloane at Wed Oct 10 09:32:03 EDT 2018

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proposed

approved

#25 by Vincenzo Librandi at Wed Sep 26 05:02:37 EDT 2018

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editing

proposed

#24 by Vincenzo Librandi at Wed Sep 26 04:40:26 EDT 2018

LINKS

Vincenzo Librandi, <a href="/A056311/b056311.txt">Table of n, a(n) for n = 1..1000</a>

#23 by Vincenzo Librandi at Wed Sep 26 04:39:40 EDT 2018

MATHEMATICA

CoefficientList[Series[12 x^3 (3 x + 1) (8 x^4 - 3 x^3 - 2 x^2 - x + 1) / ((x - 1) (4 x - 1) (3 x - 1) (2 x + 1) (2 x - 1) (3 x^2 - 1) (2 x^2 - 1)), {x, 0, 33}], x] (* Vincenzo Librandi, Sep 26 2018 *)

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proposed

editing

#22 by Jon E. Schoenfield at Tue Sep 25 23:29:54 EDT 2018

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editing

proposed

#21 by Jon E. Schoenfield at Tue Sep 25 23:29:52 EDT 2018

FORMULA

Equals A032121(n) - 4*A032120(n) + 6*A005418(n+1) - 4.

G.f.: 12*x^4*(3*x+1)*(8*x^4-3*x^3-2*x^2-x+1)/ ((x-1) * (4*x-1) * (3*x-1) * (2*x+1) * (2*x -1) * (3*x^2-1) * (2*x^2-1)) [From . - Maksym Voznyy (voznyy(AT)mail.ru), Jul 27 2009] [Corrected by R. J. Mathar, Sep 16 2009]

MATHEMATICA

k=4; Table[(StirlingS2[i, k]+StirlingS2[Ceiling[i/2], k])k!/2, {i, k, 30}] (*_ _Robert A. Russell_, Nov 25 2017 *)

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proposed

editing

#20 by Robert A. Russell at Tue Sep 25 22:03:50 EDT 2018

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editing

proposed

#19 by Robert A. Russell at Tue Sep 25 22:03:41 EDT 2018

FORMULA

a(n) = k! (S2(n,k) + S2(ceiling(n/2),k)) / 2, where k=4 is the number of colors and S2 is the Stirling subset number. - Robert A. Russell, Sep 25 2018

EXAMPLE

For n=4, the 12 rows are 12 permutations of ABCD that do not include any mutual reversals. Each of the 12 chiral pairs, such as ABCD-DCBA, is then counted just once. - Robert A. Russell, Sep 25 2018

CROSSREFS

Column 4 of A305621.

Equals (A000919 + A056455) / 2 = A000919 - A305624 = A305624 + A056455.

STATUS

approved

editing

#18 by N. J. A. Sloane at Sat Nov 25 09:35:29 EST 2017

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proposed

approved

#17 by Robert A. Russell at Sat Nov 25 08:40:15 EST 2017

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editing

proposed