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%I #25 Mar 24 2023 16:19:09
%S 0,0,6,36,132,540,2076,8316,32892,131580,524796,2099196,8390652,
%T 33562620,134225916,536903676,2147516412,8590065660,34359869436,
%U 137439477756,549756338172,2199025352700,8796095119356,35184380477436,140737496743932
%N Number of n-tuples of 4 elements excluding reverse duplicates and those consisting of repetitions of the same element only.
%C Also the number of distinct DNA or RNA sequences of length n if the reverse copies and homopolymeric oligonucleotides (i.e., repetitions of the same nucleobases: aaa..., ccc..., ggg..., and ttt... (or uuu...)) are excluded.
%H Colin Barker, <a href="/A321579/b321579.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (5,0,-20,16).
%F a(n) = (2^(n-2)*((-1)^(n+1) + 3) + 2^(2*n-1) - 4) for n > 0.
%F a(n) = A032121(n) - 4 for n > 2.
%F G.f.: 6*x^2*(8*x^2 - x - 1)/((x-1)*(2*x+1)*(2*x-1)*(4*x-1)).
%F a(n) = 5*a(n-1) - 20*a(n-3) + 16*a(n-4). - _Colin Barker_, Nov 14 2018
%e a(2) = 6 because {a,c,g,t} give six 2-tuples (duples): {a,c}, {a,g}, {a,t}, {c,g}, {c,t}, {g,t} as 4: {a,a}, {c,c}, {g,g}, {t,t} (consisting of the same element only) and 6 reverse duplicates: {c,a}, {g,a}, {t,a}, {g,c}, {t,c}, {t,g} are excluded ({c,a} is the duplicate of {a,c}, etc.), leaving 6 from 16 possible 2-tuples.
%t a[n_]:=(2^(# - 2)*((-1)^(# + 1) + 3) + 2^(2*# - 1) - 4)&/@ Range@n; a[25] (* or *)
%t CoefficientList[Series[6*(8*x^3-x^2-x)/(16*x^4-20*x^3+5*x-1), {x, 0, 20}], x]
%t LinearRecurrence[{5,0,-20,16},{0,0,6,36,132},30] (* _Harvey P. Dale_, Mar 20 2023 *)
%o (PARI) concat([0,0], Vec(6*x^2*(1 + x - 8*x^2) / ((1 - x)*(1 - 2*x)*(1 + 2*x)*(1 - 4*x)) + O(x^40))) \\ _Colin Barker_, Nov 14 2018
%Y Cf. A032121.
%K nonn,easy
%O 0,3
%A _Mikk Heidemaa_, Nov 13 2018
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