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A321579 Number of n-tuples of 4 elements excluding reverse duplicates and those consisting of repetitions of the same element only. 1
0, 0, 6, 36, 132, 540, 2076, 8316, 32892, 131580, 524796, 2099196, 8390652, 33562620, 134225916, 536903676, 2147516412, 8590065660, 34359869436, 137439477756, 549756338172, 2199025352700, 8796095119356, 35184380477436, 140737496743932 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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.

LINKS

Colin Barker, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (5,0,-20,16).

FORMULA

a(n) = (2^(n-2)*((-1)^(n+1) + 3) + 2^(2*n-1) - 4) for n > 0.

a(n) = A032121(n) - 4 for n > 2.

G.f.: 6*x^2*(8*x^2 - x - 1)/(16*x^4 - 20*x^3 + 5*x - 1).

a(n) = 5*a(n-1) - 20*a(n-3) + 16*a(n-4). - Colin Barker, Nov 14 2018

EXAMPLE

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.

MATHEMATICA

a[n_]:=(2^(# - 2)*((-1)^(# + 1) + 3) + 2^(2*# - 1) - 4)&/@ Range@n; a[25] (* or *)

CoefficientList[Series[6*(8*x^3-x^2-x)/(16*x^4-20*x^3+5*x-1), {x, 0, 20}], x]

PROG

(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

CROSSREFS

Cf. A032121.

Sequence in context: A061707 A253945 A056375 * A018214 A181478 A223841

Adjacent sequences:  A321576 A321577 A321578 * A321580 A321581 A321582

KEYWORD

nonn,easy

AUTHOR

Mikk Heidemaa, Nov 13 2018

STATUS

approved

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Last modified October 16 21:10 EDT 2019. Contains 328103 sequences. (Running on oeis4.)