%I #16 Sep 08 2022 08:46:11

%S 1,1,1,1,2,3,4,3,4,6,12,12,12,12,24,36,48,36,48,72,144,144,144,144,

%T 288,432,576,432,576,864,1728,1728,1728,1728,3456,5184,6912,5184,6912,

%U 10368,20736,20736,20736,20736,41472,62208,82944,62208,82944,124416,248832

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

%H Colin Barker, <a href="/A253852/b253852.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,0,0,0,0,0,12).

%F a(n) = 1 / a(3-n) for all n in Z.

%F a(n+10) = 12*a(n), a(n+7)*a(n) = a(n+5)*a(n+2), a(n+6)*a(n+5) = 12*a(n+1)*a(n) for all n in Z.

%F 0 = a(n)*(+a(n+1) + a(n+3)) + a(n+1)*(-a(n+4)) for all n in Z.

%F a(n) = 12^floor(n/10)*((1+0^((n-4) mod 10)+2*0^((n-5) mod 10)+2*0^((n-7) mod 10)+3*0^((n-6) mod 10)+3*0^((n-8) mod 10)+5*0^((n-9) mod 10)) mod 10). - _Wesley Ivan Hurt_, Apr 28 2015

%F G.f.: -(6*x^9+4*x^8+3*x^7+4*x^6+3*x^5+2*x^4+x^3+x^2+x+1) / (12*x^10-1). - _Colin Barker_, Apr 28 2015

%e G.f. = 1 + x + x^2 + x^3 + 2*x^4 + 3*x^5 + 4*x^6 + 3*x^7 + 4*x^8 + 6*x^9 + ...

%t a[n_] := a[n] = a[n - 4] (a[n - 3] + a[n - 1])/a[n - 3]; a[0] = a[1] = a[2] = a[3] = 1; Array[a, 50] (* or *)

%t LinearRecurrence[{0, 0, 0, 0, 0, 0, 0, 0, 0, 12}, {1, 1, 1, 2, 3, 4, 3, 4, 6, 12}, 50] (* or *)

%t CoefficientList[ Series[(6x^9 + 4x^8 + 3x^7 + 4x^6 + 3x^5 + 2x^4 + x^3 + x^2 + x + 1)/(1 - 12 x^10), {x, 0, 50}], x] (* _Robert G. Wilson v_, Apr 28 2015 *)

%o (PARI) {a(n) = my(q=n\10, r=n%10+1); 2^([0, 0, 0, 0, 1, 0, 2, 0, 2, 1][r]+2*q) * 3^([0, 0, 0, 0, 0, 1, 0, 1, 0, 1][r]+q)};

%o (PARI) Vec(-(6*x^9+4*x^8+3*x^7+4*x^6+3*x^5+2*x^4+x^3+x^2+x+1)/(12*x^10-1) + O(x^100)) \\ _Colin Barker_, Apr 28 2015

%o (Magma) I:=[1,1,1,1,2,3,4,3,4,6]; [n le 10 select I[n] else 12*Self(n-10): n in [1..100]]; // _Vincenzo Librandi_, Apr 29 2015

%K nonn,easy

%O 0,5

%A _Michael Somos_, Jan 17 2015