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A056492 Number of periodic palindromes using exactly six different symbols. 5
0, 0, 0, 0, 0, 0, 0, 0, 0, 360, 720, 7920, 15120, 103320, 191520, 1048320, 1905120, 9170280, 16435440, 72833040, 129230640, 541130040, 953029440, 3832187040, 6711344640, 26192766600, 45674188560, 174286672560, 302899156560, 1136023139160, 1969147121760 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,10

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..500

Index entries for linear recurrences with constant coefficients, signature (1,20,-20,-155,155,580,-580,-1044, 1044,720,-720).

FORMULA

a(n) = 2*A056346(n) - A056286(n).

G.f.: 360*x^10*(x+1)/((x-1)*(2*x-1)*(2*x+1)*(2*x^2-1)*(3*x^2-1)*(5*x^2-1)*(6*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=6 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) + 20*a(n-2) - 20*a(n-3) - 155*a(n-4) + 155*a(n-5) + 580*a(n-6) - 580*a(n-7) - 1044*a(n-8) + 1044*a(n-9) + 720*a(n-10) - 720*a(n-11). - Muniru A Asiru, Sep 26 2018

EXAMPLE

For example, aaabbb is not a (finite) palindrome but it is a periodic palindrome.

There are 720 permutations of the six letters used in ABACDEFEDC.  These 720 arrangements can be paired up with a half turn (e.g., ABACDEFEDC-EFEDCABACD) to arrive at the 360 arrangements for n=10.

MAPLE

with(combinat):  a:=n->(factorial(6)/2)*(Stirling2(floor((n+1)/2), 6)+Stirling2(ceil((n+1)/2), 6)): seq(a(n), n=1..35); # Muniru A Asiru, Sep 26 2018

MATHEMATICA

k = 6; 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, 20, -20, -155, 155, 580, -580, -1044, 1044, 720, -720}, Join[Table[0, {9}], {360, 720}], 40] (* Robert A. Russell, Sep 29 2018 *)

PROG

(PARI) a(n) = my(k=6); (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, 0, 0, 360, 720];; for n in [12..35] do a[n]:=a[n-1] +20*a[n-2]-20*a[n-3]-155*a[n-4]+155*a[n-5]+580*a[n-6] -580*a[n-7] -1044*a[n-8]+1044*a[n-9]+720*a[n-10]-720*a[n-11]; od; a; # Muniru A Asiru, Sep 26 2018

(MAGMA) m:=50; R<x>:=PowerSeriesRing(Integers(), m); [0, 0, 0, 0, 0, 0, 0, 0, 0] cat Coefficients(R!(360*x^10*(x+1)/((x-1)*(2*x-1)*(2*x+1)*(2*x^2-1)*(3*x^2-1)*(5*x^2-1)*(6*x^2-1)))); // G. C. Greubel, Oct 13 2018

CROSSREFS

Cf. A056286, A056346, A056457.

Column 6 of A305540.

Sequence in context: A076205 A048978 A056502 * A158308 A205738 A112536

Adjacent sequences:  A056489 A056490 A056491 * A056493 A056494 A056495

KEYWORD

nonn,easy

AUTHOR

Marks R. Nester

STATUS

approved

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Last modified September 29 07:28 EDT 2020. Contains 337425 sequences. (Running on oeis4.)