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%I #60 Feb 09 2024 12:49:02
%S 0,12,48,240,960,4032,16128,65280,261120,1047552,4190208,16773120,
%T 67092480,268419072,1073676288,4294901760,17179607040,68719214592,
%U 274876858368,1099510579200,4398042316800,17592181850112,70368727400448,281474959933440,1125899839733760
%N Number of non-palindromic n-tuples of 4 distinct elements.
%C Non-palindromic vs palindromic (DNA) sequences (e.g., {a,c,a,c} is a non-palindromic sequence but {a,c,c,a} is palindromic). Useful in bioinformatics.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (4,4,-16).
%F a(n) = 2^(n-1) * (2^(n+1) + (-1)^n - 3).
%F a(n) = 4^n - 4^ceiling(n/2) = A000302(n) - A056450(n).
%F From _Colin Barker_, Aug 12 2014: (Start)
%F a(n) = 4*a(n-1) + 4*a(n-2) - 16*a(n-3).
%F G.f.: 12*x^2 / ((2*x-1)*(2*x+1)*(4*x-1)). (End)
%e For n=2 the a(2)=12 solutions (non-palindromic 2-tuples over 4 distinct elements {a,c,g,t}) are: {a,c}, {a,g}, {a,t}, {c,a}, {c,g}, {c,f}, {g,a},{g,c}, {g,t}, {t,a}, {t,c}, {t,g}.
%t Table[2^(n-1) * (2^(n+1) + (-1)^n - 3), {n, 66}]
%t LinearRecurrence[{4,4,-16},{0,12,48},30] (* _Harvey P. Dale_, May 24 2023 *)
%o (PARI) a(n) = ((-1)^n - 3)*2^(n-1) + 4^n; \\ _Michel Marcus_, Aug 12 2014
%o (PARI) concat(0, Vec(12*x^2 / ((2*x-1)*(2*x+1)*(4*x-1)) + O(x^100))) \\ _Colin Barker_, Aug 12 2014
%Y Cf. A000302, A056450, A233411, A242278, A240437.
%K nonn,easy
%O 1,2
%A _Mikk Heidemaa_, Aug 12 2014
%E Typos in formula fixed by _Colin Barker_, Aug 12 2014
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