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A206282 a(n) = ( a(n-1) * a(n-3) + a(n-2) ) / a(n-4), a(1) = a(2) = 1, a(3) = -1, a(4) = -4.

%I A206282 #30 Mar 15 2024 12:49:08
%S A206282 1,1,-1,-4,-5,1,9,11,-4,-25,-31,9,64,79,-25,-169,-209,64,441,545,-169,
%T A206282 -1156,-1429,441,3025,3739,-1156,-7921,-9791,3025,20736,25631,-7921,
%U A206282 -54289,-67105,20736,142129,175681,-54289,-372100,-459941,142129,974169,1204139
%N A206282 a(n) = ( a(n-1) * a(n-3) + a(n-2) ) / a(n-4), a(1) = a(2) = 1, a(3) = -1, a(4) = -4.
%C A206282 This satisfies the same recurrence as Dana Scott's sequence A048736.
%H A206282 Reinhard Zumkeller, <a href="/A206282/b206282.txt">Table of n, a(n) for n = 1..5000</a>
%H A206282 <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,-2,0,0,2,0,0,1).
%F A206282 G.f.: x * (1 + x - x^2 - 2*x^3 - 3*x^4 - x^5 - x^6 - x^7) / (1 + 2*x^3 - 2*x^6 - x^9).
%F A206282 a(n) = a(-5 - n) = a(n+2) * a(n-2) - a(n+1) * a(n-1) for all n in Z.
%F A206282 a(3*n) = (-1)^n * F(n)^2, a(3*n + 1) = (-1)^n * F(n + 2)^2 where F = Fibonacci A000045.
%F A206282 a(6*n - 4) = - A110034(2*n), a(6*n - 1) = - A110035(2*n), a(3*n + 2) = (-1)^n * A126116(2*n + 3).
%e A206282 G.f. = x + x^2 - x^3 - 4*x^4 - 5*x^5 + x^6 + 9*x^7 + 11*x^8 - 4*x^9 - 25*x^10 + ...
%t A206282 CoefficientList[Series[x*(1+x)*(1-x^2)*(1+x^3)/(1-2*x^2-2*x^4-2*x^6+x^8 ), {x,0,50}], x] (* or *) RecurrenceTable[{a[n] == ( a[n-1]*a[n-3] + a[n-2] )/a[n-4], a[1] == a[2] == 1, a[3] == -1, a[4] == -4}, a, {n,1,50}] (* _G. C. Greubel_, Aug 12 2018 *)
%o A206282 (PARI) {a(n) = my(k = n\3); (-1)^k * if( n%3 == 0, fibonacci( k )^2, n%3 == 1, fibonacci( k+2 )^2, fibonacci( k ) * fibonacci( k+3 ) + fibonacci( k+1 ) * fibonacci( k+2 ))};
%o A206282 (PARI) x='x+O('x^30); Vec(x*(1+x)*(1-x^2)*(1+x^3)/(1-2*x^2-2*x^4 -2*x^6 +x^8 )) \\ _G. C. Greubel_, Aug 12 2018
%o A206282 (Haskell)
%o A206282 a206282 n = a206282_list !! (n-1)
%o A206282 a206282_list = 1 : 1 : -1 : -4 :
%o A206282    zipWith div
%o A206282      (zipWith (+)
%o A206282        (zipWith (*) (drop 3 a206282_list)
%o A206282                     (drop 1 a206282_list))
%o A206282        (drop 2 a206282_list))
%o A206282      a206282_list
%o A206282 -- Same program as in A048736, see comment.
%o A206282 -- _Reinhard Zumkeller_, Feb 08 2012
%o A206282 (Magma) I:=[1,1,-1,-4]; [n le 4 select I[n] else (Self(n-1)*Self(n-3) + Self(n-2))/Self(n-4): n in [1..30]]; // _G. C. Greubel_, Aug 12 2018
%Y A206282 Cf. A000045, A048736, A110034, A110035, A126116.
%K A206282 sign,easy
%O A206282 1,4
%A A206282 _Michael Somos_, Feb 05 2012