cp's OEIS Frontend

A056454 Number of palindromes of length n using exactly three different symbols.

%I A056454 #41 Feb 12 2025 14:35:08
%S A056454 0,0,0,0,6,6,36,36,150,150,540,540,1806,1806,5796,5796,18150,18150,
%T A056454 55980,55980,171006,171006,519156,519156,1569750,1569750,4733820,
%U A056454 4733820,14250606,14250606,42850116,42850116,128746950,128746950,386634060,386634060,1160688606
%N A056454 Number of palindromes of length n using exactly three different symbols.
%D A056454 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.]
%H A056454 Vincenzo Librandi, <a href="/A056454/b056454.txt">Table of n, a(n) for n = 1..1000</a>
%H A056454 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,5,-5,-6,6).
%F A056454 a(n) = 3! * Stirling2( [(n+1)/2], 3).
%F A056454 G.f.: 6*x^5/((1-x)*(1-2*x^2)*(1-3*x^2)). - _Colin Barker_, May 05 2012
%F A056454 G.f.: k!(x^(2k-1)+x^(2k))/Product_{i=1..k}(1-i*x^2), where k=3 is the number of symbols. - _Robert A. Russell_, Sep 25 2018
%p A056454 A056454:= n-> 3!*Stirling2(floor((n+1)/2),3); # (C. Ronaldo)
%t A056454 LinearRecurrence[{1,5,-5,-6,6},{0,0,0,0,6},40] (* _Harvey P. Dale_, Sep 02 2016 *)
%t A056454 k=3; Table[k! StirlingS2[Ceiling[n/2],k],{n,1,30}] (* _Robert A. Russell_, Sep 25 2018 *)
%o A056454 (PARI) a(n) = 3!*stirling((n+1)\2, 3, 2); \\ _Altug Alkan_, Sep 25 2018
%o A056454 (Magma) [StirlingSecond((n+1) div 2, 3)*Factorial(3): n in [1..40]]; // _Vincenzo Librandi_, Sep 26 2018
%Y A056454 Cf. A056449, A001117.
%K A056454 nonn,easy
%O A056454 1,5
%A A056454 _Marks R. Nester_