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

Author

Mikk Heidemaa, Nov 13 2018

Keywords

• nonn
• easy

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.

Examples

			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.
		

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).

Crossrefs

Cf. A032121.

Programs

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 *)

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

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