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A056489 Number of periodic palindromes using exactly three different symbols. 5
0, 0, 0, 3, 6, 21, 36, 93, 150, 345, 540, 1173, 1806, 3801, 5796, 11973, 18150, 37065, 55980, 113493, 171006, 345081, 519156, 1044453, 1569750, 3151785, 4733820, 9492213, 14250606, 28550361, 42850116, 85798533, 128746950, 257690505, 386634060, 773661333 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

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

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

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (1,5,-5,-6,6).

FORMULA

a(n) = 2 * A056343(n) - A056283(n).

G.f.: 3*x^4*(1+x)/((1-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=3 different colors used and where S2(n,k) is the Stirling subset number A008277. - Robert A. Russell, Jun 05 2018

EXAMPLE

For n=4, the three arrangements are ABAC, ABCB, and ACBC.

For n=5, the six arrangements are AABCB, AACBC, ABACC, ABBAC, ABCCB, and ACBBC.

MAPLE

seq(coeff(series(3*x^4*(1+x)/((1-x)*(1-2*x^2)*(1-3*x^2)), x, n+1), x, n), n = 1..40); # Muniru A Asiru, Sep 28 2018

MATHEMATICA

k = 3; 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, 5, -5, -6, 6}, {0, 0, 0, 3, 6}, 80] (* Vincenzo Librandi, Sep 27 2018 *)

PROG

(PARI) a(n) = my(k=3); (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, 3, 6];; for n in [6..40] do a[n]:=a[n-1]+5*a[n-2]-5*a[n-3]-6*a[n-4]+6*a[n-5]; od; a; # Muniru A Asiru, Sep 28 2018

CROSSREFS

Cf. A056454.

Column 3 of A305540.

Sequence in context: A270510 A300891 A056499 * A015649 A297189 A076102

Adjacent sequences:  A056486 A056487 A056488 * A056490 A056491 A056492

KEYWORD

nonn,easy

AUTHOR

Marks R. Nester

STATUS

approved

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Last modified August 18 01:04 EDT 2019. Contains 326059 sequences. (Running on oeis4.)