%I #36 Feb 13 2026 10:02:47

%S 1,1,1,3,1,1,1,3,1,1,1,3,1,1,1,3,1,1,1,3,1,1,1,3,1,1,1,3,1,1,1,3,1,1,

%T 1,3,1,1,1,3,1,1,1,3,1,1,1,3,1,1,1,3,1,1,1,3,1,1,1,3,1,1,1,3,1,1,1,3,

%U 1,1,1,3,1,1,1,3,1,1,1,3,1,1,1,3,1,1,1,3,1,1,1,3,1,1,1,3,1,1,1,3,1,1,1,3,1

%N a(n) = gcd(Fibonacci(n+5), Fibonacci(n+1)).

%C From _Klaus Brockhaus_, May 30 2010: (Start)

%C Periodic sequence: Repeat [1, 1, 1, 3].

%C Continued fraction expansion of (9+sqrt(165))/14.

%C Decimal expansion of 371/3333. (End)

%C Final nonzero digit of n^n in base 4. - _José María Grau Ribas_, Jan 19 2012

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,1).

%F G.f.: (1+x+x^2+3*x^3)/(1-x^4).

%F a(n) = 3/2-sin(Pi*n/2)-cos(Pi*n)/2.

%F From _Klaus Brockhaus_, May 30 2010: (Start)

%F a(n) = a(n-4) for n > 3; a(0) = a(1) = a(2) = 1, a(3) = 3.

%F a(n) = (3-(-1)^n+(1-(-1)^n)*i*i^n)/2 where i = sqrt(-1). (End)

%F a(n) = 1 + 2*0^mod(n+1, 4). - _Wesley Ivan Hurt_, Oct 23 2014

%F E.g.f.: cosh(x) - sin(x) + 2*sinh(x). - _Amiram Eldar_, Feb 13 2026

%p A093148:=n->1+2*0^(n+1 mod 4): seq(A093148(n), n=0..100); # _Wesley Ivan Hurt_, Oct 23 2014

%t f[n_] := Switch[Mod[n, 4], 0, 1, 1, 1, 2, 1, 3, 3]; Array[f, 105, 0] (* _Robert G. Wilson v_, Jan 23 2012 *)

%t PadRight[{},120,{1,1,1,3}] (* _Harvey P. Dale_, Sep 03 2021 *)

%o (Magma) [1+2*0^((n+1) mod 4) : n in [0..100]]; // _Wesley Ivan Hurt_, Oct 23 2014

%o (PARI) a(n)=if(n%4==3,3,1) \\ _Charles R Greathouse IV_, Oct 07 2015

%Y Cf. A000045, A061347, A167816, A177704, A178591.

%K easy,nonn

%O 0,4

%A _Paul Barry_, Apr 02 2004