A056489 Number of periodic palindromes using exactly three different symbols.

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

Offset: 1

Author

Marks R. Nester

Keywords

• nonn
• easy

Comments

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

Examples

			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.
		

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).

Crossrefs

Cf. A056454.

Column 3 of A305540.

Programs

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

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 *)

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

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