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A333599 a(n) = Fibonacci(n) * Fibonacci(n+1) mod Fibonacci(n+2).

%I A333599 #41 May 22 2020 09:46:41
%S A333599 0,1,2,1,7,1,20,1,54,1,143,1,376,1,986,1,2583,1,6764,1,17710,1,46367,
%T A333599 1,121392,1,317810,1,832039,1,2178308,1,5702886,1,14930351,1,39088168,
%U A333599 1,102334154,1,267914295,1,701408732,1,1836311902,1,4807526975,1,12586269024
%N A333599 a(n) = Fibonacci(n) * Fibonacci(n+1) mod Fibonacci(n+2).
%H A333599 Colin Barker, <a href="/A333599/b333599.txt">Table of n, a(n) for n = 0..1000</a>
%H A333599 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/dOcagnesIdentity.html">d'Ocagne's Identity</a>.
%H A333599 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (-1,3,3,-1,-1).
%F A333599 a(2n+1) = 1, and a(2n) = F(2n+2) - 1, and lim(a(2n+2)/a(2n)) = phi^2 by d'Ocagne's identity.
%F A333599 a(n) = F(n) * F(n+1) mod (F(n) + F(n+1)) since F(n+2) := F(n+1) + F(n).
%F A333599 From _Colin Barker_, Mar 28 2020: (Start)
%F A333599 G.f.: x*(1 + 3*x - x^3) / ((1 + x)*(1 + x - x^2)*(1 - x - x^2)).
%F A333599 a(n) = -a(n-1) + 3*a(n-2) + 3*a(n-3) - a(n-4) - a(n-5) for n>4.
%F A333599 (End)
%e A333599 a(0) = 0*1 mod 1 = 0;
%e A333599 a(1) = 1*1 mod 2 = 1;
%e A333599 a(2) = 1*2 mod 3 = 2;
%e A333599 a(3) = 2*3 mod 5 = 1;
%e A333599 a(4) = 3*5 mod 8 = 7.
%t A333599 With[{f = Fibonacci}, Table[Mod[f[n] * f[n+1], f[n+2]], {n, 0, 50}]] (* _Amiram Eldar_, Mar 28 2020 *)
%o A333599 (Python)
%o A333599 def a(n):
%o A333599     f1 = 0
%o A333599     f2 = 1
%o A333599     for i in range(n):
%o A333599         f = f1 + f2
%o A333599         f1 = f2
%o A333599         f2 = f
%o A333599     return (f1 * f2) % (f1 + f2)
%o A333599 (PARI) a(n) = if (n % 2, 1, fibonacci(n+2) - 1); \\ _Michel Marcus_, Mar 29 2020
%o A333599 (PARI) concat(0, Vec(x*(1 + 3*x - x^3) / ((1 + x)*(1 + x - x^2)*(1 - x - x^2)) + O(x^45))) \\ _Colin Barker_, Mar 29 2020
%Y A333599 Equals A035508 interleaved with A000012.
%Y A333599 Cf. A182126, A000045, A022461, A072565, A022462.
%K A333599 nonn,easy
%O A333599 0,3
%A A333599 _Adnan Baysal_, Mar 28 2020