%I #31 Oct 09 2023 02:20:40

%S 3,12,15,2,21,24,9,30,33,12,39,42,5,48,51,18,57,60,21,66,69,8,75,78,

%T 27,84,87,30,93,96,11,102,105,36,111,114,39,120,123,14,129,132,45,138,

%U 141,48,147,150,17,156,159,54,165,168,57,174,177,20,183,186,63

%N a(n) = 3*n/gcd(3*n, n+3), n >= 3.

%C This is the third column sequence (m = 3) of the triangle A221918.

%H <a href="/index/Rec#order_18">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,-1).

%F a(n) = A221918(n,3) = numerator(n*3/(n+3)), n >= 3.

%F a(n) = 3*n/gcd(3*n,n+3), n >= 3.

%F a(n) = 3*n/gcd(9,n+3), n >= 3, (because gcd(n+3,3*n) = gcd(n+3,3*n - 3*(n+3)) = gcd(n+3,-3^2) = gcd(n+3,9)).

%F G.f.: -x^3*(6*x^17 + 3*x^16 - 3*x^14 - 6*x^13 - x^12 - 12*x^11 - 15*x^10 - 6*x^9 - 33*x^8 - 30*x^7 - 9*x^6 - 24*x^5 - 21*x^4 - 2*x^3 - 15*x^2 - 12*x - 3) / ((x-1)^2*(x^2 + x + 1)^2*(x^6 + x^3 + 1)^2). - _Colin Barker_, Feb 25 2013

%F Sum_{k=3..n} a(k) ~ (61/54) * n^2. - _Amiram Eldar_, Oct 09 2023

%e a(6) = numerator(18/9) = numerator(2) = 2 = 18/gcd(18,9) = 18/9 = 18/gcd(9,9) = 18/9.

%t a[n_] := 3*n/GCD[3*n, n+3]; Array[a, 63, 3] (* _Amiram Eldar_, Oct 09 2023 *)

%o (PARI) a(n)=3*n/gcd(3*n,n+3) \\ _Charles R Greathouse IV_, Apr 18 2013

%Y Cf. A221918, A000027 (m=1), A145979(m=2), A221921 (m=4), A222463 (m=5).

%K nonn,easy

%O 3,1

%A _Wolfdieter Lang_, Feb 21 2013