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A056490 Number of periodic palindromes using exactly four different symbols. 5
0, 0, 0, 0, 0, 12, 24, 132, 240, 900, 1560, 4980, 8400, 24612, 40824, 113652, 186480, 502500, 818520, 2158260, 3498000, 9087012, 14676024, 37728372, 60780720, 155091300, 249401880, 632972340, 1016542800, 2569858212, 4123173624, 10393634292, 16664094960 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,6

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..700

Index entries for linear recurrences with constant coefficients, signature (1,9,-9,-26,26,24,-24).

FORMULA

a(n) = 2*A056344(n) - A056284(n).

G.f.: 12*x^6*(1+x)/((1-x)*(1-2*x)*(1+2*x)*(1-2*x^2)*(1-3*x^2)). - Colin Barker, May 06 2012

a(n) = (k!/2)*(S2(floor((n+1)/2),k) + S2(ceiling((n+1)/2),k)), with k=4 different colors used and where S2(n,k) is the Stirling subset number A008277. - Robert A. Russell, Jun 05 2018

EXAMPLE

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

For n=6, the 12 arrangements are ABACDC, ABADCD, ACABDB, ACADBD, ADABCB, ADACBC, ABCDCB, ABDCDB, ACBDBC, ACDBDC, ADBCBD, and ADCBCD.

MAPLE

a:=n->(factorial(4)/2)*(Stirling2(floor((n+1)/2), 4)+Stirling2(ceil((n+1)/2), 4)): seq(a(n), n=1..35); # Muniru A Asiru, Sep 26 2018

MATHEMATICA

k = 4; 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, 9, -9, -26, 26, 24, -24}, {0, 0, 0, 0, 0, 12, 24}, 40] (* Robert A. Russell, Sep 29 2018 *)

PROG

(PARI) a(n) = my(k=4); (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, 12, 24];; for n in [8..35] do a[n]:=a[n-1]+9*a[n-2]-9*a[n-3]-26*a[n-4]+26*a[n-5]+24*a[n-6]-24*a[n-7]; od; a; # Muniru A Asiru, Sep 26 2018

(MAGMA) m:=50; R<x>:=PowerSeriesRing(Integers(), m); [0, 0, 0, 0, 0] cat Coefficients(R!(12*x^6*(1+x)/((1-x)*(1-2*x)*(1+2*x)*(1-2*x^2)*(1-3*x^2)))); // G. C. Greubel, Oct 13 2018

CROSSREFS

Cf. A056284, A056344, A056455.

Column 4 of A305540.

Sequence in context: A290304 A289335 A056500 * A130163 A002167 A154268

Adjacent sequences:  A056487 A056488 A056489 * A056491 A056492 A056493

KEYWORD

nonn,easy

AUTHOR

Marks R. Nester

STATUS

approved

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Last modified August 23 22:45 EDT 2019. Contains 326254 sequences. (Running on oeis4.)