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%I #45 Sep 08 2022 08:45:01
%S 0,0,0,0,0,0,0,0,0,360,720,7920,15120,103320,191520,1048320,1905120,
%T 9170280,16435440,72833040,129230640,541130040,953029440,3832187040,
%U 6711344640,26192766600,45674188560,174286672560,302899156560,1136023139160,1969147121760
%N Number of periodic palindromes using exactly six different symbols.
%D 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 Muniru A Asiru, <a href="/A056492/b056492.txt">Table of n, a(n) for n = 1..500</a>
%H <a href="/index/Rec#order_11">Index entries for linear recurrences with constant coefficients</a>, signature (1,20,-20,-155,155,580,-580,-1044, 1044,720,-720).
%F a(n) = 2*A056346(n) - A056286(n).
%F 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
%F 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
%F 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
%e For example, aaabbb is not a (finite) palindrome but it is a periodic palindrome.
%e 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.
%p 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
%t 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 *)
%t 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 *)
%o (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
%o (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
%o (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
%Y Cf. A056286, A056346, A056457.
%Y Column 6 of A305540.
%K nonn,easy
%O 1,10
%A _Marks R. Nester_
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