|
%I #36 Feb 22 2023 01:49:16
%S 0,22,66,132,220,330,462,616,792,990,1210,1452,1716,2002,2310,2640,
%T 2992,3366,3762,4180,4620,5082,5566,6072,6600,7150,7722,8316,8932,
%U 9570,10230,10912,11616,12342,13090,13860,14652,15466,16302,17160,18040,18942,19866,20812
%N a(n) = 11*n*(n+1).
%H Vincenzo Librandi, <a href="/A164136/b164136.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F a(n) = 22*A000217(n) = 11*A002378(n).
%F a(n) = 22*n + a(n-1) with n>0, a(0)=0. - _Vincenzo Librandi_, Nov 30 2010
%F G.f.: 22*x/(1-x)^3. - _Vincenzo Librandi_, Sep 12 2013
%F a(n) = 3*a(n-1) -3*a(n-2) +a(n-3). - _Vincenzo Librandi_, Sep 12 2013
%F E.g.f.: 11*x*(2 + x)*exp(x). - _G. C. Greubel_, Sep 12 2017
%F From _Amiram Eldar_, Feb 22 2023: (Start)
%F Sum_{n>=1} 1/a(n) = 1/11.
%F Sum_{n>=1} (-1)^(n+1)/a(n) = (2*log(2) - 1)/11.
%F Product_{n>=1} (1 - 1/a(n)) = -(11/Pi)*cos(sqrt(15/11)*Pi/2).
%F Product_{n>=1} (1 + 1/a(n)) = (11/Pi)*cos(sqrt(7/11)*Pi/2). (End)
%t CoefficientList[Series[(22 x) / (1 - x)^3, {x, 0, 40}], x] (* _Vincenzo Librandi_, Sep 12 2013 *)
%o (Magma) [11*n*(n+1): n in [0..40]]; // _Vincenzo Librandi_, Sep 12 2013
%o (PARI) a(n)=11*n*(n+1) \\ _Charles R Greathouse IV_, Jun 17 2017
%Y Cf. A000217, A002378.
%K nonn,easy
%O 0,2
%A _Vincenzo Librandi_, Aug 11 2009
%E Offset corrected by _R. J. Mathar_, Aug 21 2009
|
