A321579 Number of n-tuples of 4 elements excluding reverse duplicates and those consisting of repetitions of the same element only.

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

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)/((x-1)*(2*x+1)*(2*x-1)*(4*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]
LinearRecurrence[{5, 0, -20, 16}, {0, 0, 6, 36, 132}, 30] (* Harvey P. Dale, Mar 20 2023 *)

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: A331576 A056375 A360588 * A364429 A018214 A181478

Adjacent sequences: A321576 A321577 A321578 * A321580 A321581 A321582

Keyword

nonn,easy

Author

Mikk Heidemaa, Nov 13 2018

Status

approved