A078530 Bilinear recursive sequence.
0, 3, 1, 1, 1, 1, 2, 3, 9, 27, 81, 729, 0, 59049, -531441, 14348907, -387420489, 10460353203, -564859072962, 22876792454961, -1853020188851841, 150094635296999121, -12157665459056928801, 2954312706550833698643, 0
Offset: 0
Programs
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Mathematica
a[ n_] := With[{m = Mod[n, 12]}, Sign[m] * 2^Boole[m==6] * (-1)^(Mod[Floor[n/12], 2]*(n-1)) * 3^(Boole[m==0] + Floor[(n-4)^2/8])]; (* Michael Somos, Dec 10 2023 *) -
PARI
{a(n) = sign(n%12) * (1 + (n%12==6)) * (-1)^(n\12%2 * (n-1)) * 3^((n%12==0) + (n-4)^2\8)};
Formula
a(n) * a(n-8) = 81 * (a(n-2)*a(n-6) - 2*a(n-4)^2).
0 = a(n) * a(n-5) + 3 * a(n-1) * a(n-4) - 9 * a(n-2)*a(n-3).
a(12*n) = 0.
a(2*n+1) = a(-2*n+7) = a(4*n+2)/(81^(n-1)*(a(2*n-1)*a(2*n+2)^2 - a(2*n+3)*a(2*n)^2)) for all n in Z. - Michael Somos, Dec 10 2023
a(n+12) = -(-27)^(n+2) * a(n) for all n in Z. - Michael Somos, Dec 11 2023