cp's OEIS Frontend

A320524 Number of chiral pairs of a row of n colors using 6 or fewer colors.

%I A320524 #11 Sep 08 2022 08:46:23
%S A320524 0,15,90,630,3780,23220,139320,839160,5034960,30229200,181375200,
%T A320524 1088367840,6530207040,39181942080,235091652480,1410554113920,
%U A320524 8463324683520,50779973295360,304679839772160,1828079189798400,10968475138790400,65810851739735040,394865110438410240,2369190668072417280,14215144008434503680,85290864083258757120
%N A320524 Number of chiral pairs of a row of n colors using 6 or fewer colors.
%C A320524 A chiral row is different from its reverse.
%H A320524 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (6,6,-36).
%F A320524 a(n) = (k^n - k^ceiling(n/2)) / 2, where k=6 is maximum number of colors.
%F A320524 G.f.: k*x^2*(k-1) / (2*(1-k*x)*(1-k*x^2)), where k=6.
%F A320524 a(n) = (A000400(n) - A056452(n)) / 2 = A000400(n) - A056308(n) = A056308(n) - A056452(n).
%e A320524 For a(2)=15, the chiral pairs are the fifteen combinations of six colors taken two at a time, e.g., AB-BA.
%t A320524 k = 6; Table[(k^n - k^Ceiling[n/2])/2, {n, 1, 30}]
%t A320524 LinearRecurrence[{6, 6, -36}, {0, 15, 90}, 30]
%o A320524 (Magma) [(6^n - 6^Ceiling(n / 2)) / 2: n in [1..25]]; // _Vincenzo Librandi_, Oct 15 2018
%o A320524 (PARI) m=40; v=concat([0,15,90], vector(m-3)); for(n=4, m, v[n] = 6*v[n-1] +6*v[n-2] -36*v[n-3]); v \\ _G. C. Greubel_, Oct 17 2018
%Y A320524 Column 6 of A293500.
%Y A320524 Cf. A000400 (oriented), A056308 (unoriented), A056452 (achiral).
%K A320524 easy,nonn
%O A320524 1,2
%A A320524 _Robert A. Russell_, Oct 14 2018