A217366 - OEIS

%I #36 Sep 14 2022 19:36:08

%S 0,7,2,27,5,55,9,91,14,135,20,187,27,247,35,315,44,391,54,475,65,567,

%T 77,667,90,775,104,891,119,1015,135,1147,152,1287,170,1435,189,1591,

%U 209,1755,230,1927,252,2107,275,2295,299,2491,324,2695,350,2907,377

%N a(n) = ((n+6) / gcd(n+6,4)) * (n / gcd(n,4)).

%C The 6th sequence (p=6) of the family A060819(n)*A060819(n+p).

%H Vincenzo Librandi, <a href="/A217366/b217366.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,3,0,-3,0,1).

%F a(n) = 8^(n mod 2 - 1)*n*(n + 6).

%F G.f.: x*(7 + 2*x + 6*x^2 - x^3 - 5*x^4)/(1 - x^2)^3. - _Bruno Berselli_, Oct 01 2012

%F From _Colin Barker_, Jan 27 2016: (Start)

%F a(n) = (9 - 7*(-1)^n)*n*(n + 6)/16.

%F a(n) = (n^2 + 6*n)/8 for n even.

%F a(n) = n^2 + 6*n for n odd. (End)

%F Sum_{n>=1} 1/a(n) = 133/90. - _Amiram Eldar_, Aug 12 2022

%t a[n_] := 8^(Mod[n, 2] - 1)*n*(n + 6); Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, Oct 01 2012 *)

%t CoefficientList[Series[x*(7 + 2*x + 6*x^2 - x^3 - 5*x^4)/(1 - x^2)^3, {x, 0, 40}], x] (* _Vincenzo Librandi_, Dec 15 2012 *)

%t LinearRecurrence[{0,3,0,-3,0,1},{0,7,2,27,5,55},60] (* _Harvey P. Dale_, Sep 14 2022 *)

%o (PARI) concat(0, Vec(x*(7+2*x+6*x^2-x^3-5*x^4)/((1-x)^3*(1+x)^3) + O(x^100))) \\ _Colin Barker_, Jan 27 2016

%o (PARI) vector(50, n, n--; (9-7*(-1)^n)/16*n*(n+6)) \\ _G. C. Greubel_, Sep 21 2018

%o (Magma) [(9-7*(-1)^n)/16*n*(n+6): n in [0..50]]; // _G. C. Greubel_, Sep 21 2018

%Y Cf. A060819, A181318.

%K nonn,easy

%O 0,2

%A _Jean-François Alcover_, Oct 01 2012