A056491 Number of periodic palindromes using exactly five different symbols.

0, 0, 0, 0, 0, 0, 0, 60, 120, 960, 1800, 9300, 16800, 71400, 126000, 480060, 834120, 2968560, 5103000, 17355300, 29607600, 97567800, 165528000, 533274060, 901020120, 2855012160, 4809004200, 15050517300, 25292030400, 78417448200, 131542866000, 404936532060

Offset

1,8

References

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]

Links

Muniru A Asiru, Table of n, a(n) for n = 1..600

Index entries for linear recurrences with constant coefficients, signature (1,14,-14,-71,71,154,-154,-120, 120).

Formula

a(n) = 2*A056345(n) - A056285(n).

G.f.: -60*x^8*(x+1)/((x-1)*(2*x-1)*(2*x+1)*(2*x^2-1)*(3*x^2-1)*(5*x^2-1)). - Colin Barker, Jul 08 2012

a(n) = (k!/2)*(S2(floor((n+1)/2),k) + S2(ceiling((n+1)/2),k)), with k=5 different colors used and where S2(n,k) is the Stirling subset number A008277. - Robert A. Russell, Jun 05 2018

a(n) = a(n-1) + 14*a(n-2) - 14*a(n-3) - 71*a(n-4) + 71*a(n-5) + 154*a(n-6) - 154*a(n-7) - 120*a(n-8) + 120*a(n-9). - Muniru A Asiru, Sep 26 2018

Example

For example, aaabbb is not a (finite) palindrome but it is a periodic palindrome.
There are 120 permutations of the five letters used in ABACDEDC.  These 120 arrangements can be paired up with a half turn (e.g., ABACDEDC-DEDCABAC) to arrive at the 60 arrangements for n=8. - Robert A. Russell, Sep 26 2018

Maple

with(combinat):  a:=n->(factorial(5)/2)*(Stirling2(floor((n+1)/2), 5)+Stirling2(ceil((n+1)/2), 5)): seq(a(n), n=1..35); # Muniru A Asiru, Sep 26 2018

Mathematica

k = 5; Table[(k!/2) (StirlingS2[Floor[(n + 1)/2], k] +
StirlingS2[Ceiling[(n + 1)/2], k]), {n, 1, 40}] (* Robert A. Russell, Jun 05 2018 *)
LinearRecurrence[{1, 14, -14, -71, 71, 154, -154, -120, 120}, {0, 0,
0, 0, 0, 0, 0, 60, 120}, 40] (* Robert A. Russell, Sep 29 2018 *)

Prog

(PARI) a(n) = my(k=5); (k!/2)*(stirling(floor((n+1)/2), k, 2) + stirling(ceil((n+1)/2), k, 2)); \\ Michel Marcus, Jun 05 2018
(GAP) a:=[0, 0, 0, 0, 0, 0, 0, 60, 120];; for n in [10..35] do a[n]:=a[n-1]+14*a[n-2]-14*a[n-3]-71*a[n-4]+71*a[n-5]+154*a[n-6]-154*a[n-7]-120*a[n-8]+120*a[n-9]; od; a; # Muniru A Asiru, Sep 26 2018
(Magma) m:=50; R<x>:=PowerSeriesRing(Integers(), m); [0, 0, 0, 0, 0, 0, 0] cat Coefficients(R!(-60*x^8*(x+1)/((x-1)*(2*x-1)*(2*x+1)*(2*x^2-1)*(3*x^2-1)*(5*x^2-1)))); // G. C. Greubel, Oct 13 2018

Crossrefs

Cf. A056456.

Column 5 of A305540.

Sequence in context: A275339 A049058 A056501 * A044247 A044628 A181190

Adjacent sequences: A056488 A056489 A056490 * A056492 A056493 A056494

Keyword

nonn,easy

Author

Marks R. Nester

Status

approved