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A056491 Number of periodic palindromes using exactly five different symbols. 5
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified August 18 22:26 EDT 2019. Contains 326109 sequences. (Running on oeis4.)