%I #12 Sep 08 2022 08:46:09

%S 0,-1,1,1,1,0,1,1,1,3,4,-5,1,-7,9,8,25,-23,49,87,16,295,529,-903,841,

%T -1256,3481,-1495,16641,-44341,98596,217651,4225,1058961,2337841,

%U -5106896,13608721,5415345,67387681,-173830481,264517696,-2288275633,6941055969

%N a(n) = (a(n-1) * a(n-3) + (-1)^n * a(n-2)^2) / a(n-4), with a(0) = 0, a(1) = -1, a(2) = a(3) = a(4) = 1, a(9) = 3.

%H Reinhard Zumkeller, <a href="/A247369/b247369.txt">Table of n, a(n) for n = 0..400</a>

%F 0 = a(n)*a(n+9) + a(n+1)*a(n+8) + a(n+3)*a(n+6) + a(n+4)*a(n+5) for all n in Z.

%F a(n) = a(-n), a(2*n) = A006769(n)^2 for all n in Z.

%t Join[{0, -1, 1, 1, 1, 0, 1, 1, 1}, RecurrenceTable[{a[9]==3, a[10]==4, a[11]==-5, a[12]==1, a[n]==(a[n-1]a[n-3] + (-1)^n a[n-2]^2)/a[n-4]}, a, {n, 9, 30}]] (* _G. C. Greubel_, Aug 05 2018 *)

%o (PARI) {a(n) = my(A = [-1, 1, 1, 1]); n=abs(n); if( n==0, 0, if( n<5, A[n], A = concat(A, vector(n-4)); for(k=5, n, A[k] = if( k==9, 3, (A[k-1] * A[k-3] + (-1)^k * A[k-2]^2) / A[k-4])); A[n]))};

%o (Haskell)

%o a247369 n = a247369_list !! n

%o a247369_list = [0, -1, 1, 1, 1, 0] ++ xs where

%o xs = [1, 1, 1, 3] ++ zipWith (flip div) xs (zipWith (+)

%o (zipWith (*) (tail xs) (drop 3 xs))

%o (zipWith (*) (cycle [1, -1]) (map (^ 2) $ drop 2 xs)))

%o -- _Reinhard Zumkeller_, Sep 15 2014

%o (Magma) I:=[3, 4, -5, 1]; [0, -1, 1, 1, 1, 0, 1, 1, 1] cat [n le 4 select I[n] else ( Self(n-1)*Self(n-3) + (-1)^n*Self(n-2)^2 )/Self(n-4): n in [1..30]]; // _G. C. Greubel_, Aug 05 2018

%Y Cf. A006769.

%K sign

%O 0,10

%A _Michael Somos_, Sep 14 2014