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%I #28 Feb 22 2023 01:49:35
%S 0,18,54,108,180,270,378,504,648,810,990,1188,1404,1638,1890,2160,
%T 2448,2754,3078,3420,3780,4158,4554,4968,5400,5850,6318,6804,7308,
%U 7830,8370,8928,9504,10098,10710,11340,11988,12654,13338,14040,14760,15498,16254,17028
%N a(n) = 9*n*(n+1).
%C 18 times the n-th triangular number.
%C Numbers of the form 36*m^2 + 18*m, where m = 0,-1,1,-2,2,-3,3,... - _Bruno Berselli_, Apr 07 2013
%H Vincenzo Librandi, <a href="/A163758/b163758.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 G.f.: 18*x/(1-x)^3.
%F a(n) = 18*A000217(n) = 9*A002378(n).
%F E.g.f.: 9*x*(x + 2)*exp(x). - _G. C. Greubel_, Aug 02 2017
%F From _Amiram Eldar_, Feb 22 2023: (Start)
%F Sum_{n>=1} 1/a(n) = 1/9.
%F Sum_{n>=1} (-1)^(n+1)/a(n) = (2*log(2) - 1)/9.
%F Product_{n>=1} (1 - 1/a(n)) = -(9/Pi)*cos(sqrt(13)*Pi/6).
%F Product_{n>=1} (1 + 1/a(n)) = (9/Pi)*cos(sqrt(5)*Pi/6). (End)
%t Table[9 n (n + 1), {n, 0, 100}] (* _Vladimir Joseph Stephan Orlovsky_, Jun 26 2011 *)
%o (Magma) [0] cat [36*m^2+18*m where m is n*t: t in [-1,1], n in [1..20]]; // _Bruno Berselli_, Apr 07 2013
%o (PARI) a(n)=9*n*(n+1) \\ _Charles R Greathouse IV_, Jun 17 2017
%Y Cf. A000217, A002378.
%K nonn,easy
%O 0,2
%A _Vincenzo Librandi_, Aug 03 2009
%E a(35) inserted by _R. J. Mathar_, Aug 06 2009
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