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
KEYWORD
nonn,easy
AUTHOR
Mikk Heidemaa, Nov 13 2018
STATUS
approved