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A173307 a(n) = 13*n*(n+1).

%I A173307 #35 Dec 14 2024 11:01:55
%S A173307 0,26,78,156,260,390,546,728,936,1170,1430,1716,2028,2366,2730,3120,
%T A173307 3536,3978,4446,4940,5460,6006,6578,7176,7800,8450,9126,9828,10556,
%U A173307 11310,12090,12896,13728,14586,15470,16380,17316,18278,19266,20280,21320,22386,23478,24596
%N A173307 a(n) = 13*n*(n+1).
%H A173307 Vincenzo Librandi, <a href="/A173307/b173307.txt">Table of n, a(n) for n = 0..1000</a>
%H A173307 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A173307 a(n) = 26*A000217(n).
%F A173307 From _Vincenzo Librandi_, Sep 28 2013: (Start)
%F A173307 G.f.: 26*x/(1-x)^3.
%F A173307 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). (End)
%F A173307 From _Amiram Eldar_, Feb 22 2023: (Start)
%F A173307 Sum_{n>=1} 1/a(n) = 1/13.
%F A173307 Sum_{n>=1} (-1)^(n+1)/a(n) = (2*log(2) - 1)/13.
%F A173307 Product_{n>=1} (1 - 1/a(n)) = -(13/Pi)*cos(sqrt(17/13)*Pi/2).
%F A173307 Product_{n>=1} (1 + 1/a(n)) = (13/Pi)*cos(3*Pi/(2*sqrt(13))). (End)
%F A173307 From _Elmo R. Oliveira_, Dec 14 2024: (Start)
%F A173307 E.g.f.: 13*exp(x)*x*(2 + x).
%F A173307 a(n) = 13*A002378(n) = 2*A152741(n). (End)
%t A173307 Table[13 n (n + 1), {n, 0, 50}] (* or *) CoefficientList[Series[26 x/(1 - x)^3, {x, 0, 50}], x] (* _Vincenzo Librandi_, Sep 28 2013 *)
%t A173307 LinearRecurrence[{3,-3,1},{0,26,78},50] (* _Harvey P. Dale_, Apr 08 2014 *)
%o A173307 (Magma) [13*n*(n+1): n in [0..40]]; // _Vincenzo Librandi_, Sep 28 2013
%o A173307 (Magma) I:=[0, 26, 78]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..40]]; // _Vincenzo Librandi_, Sep 28 2013
%o A173307 (PARI) a(n)=13*n*(n+1) \\ _Charles R Greathouse IV_, Jun 17 2017
%Y A173307 Cf. A000217, A002378, A152741, A262221.
%K A173307 nonn,easy
%O A173307 0,2
%A A173307 _Vincenzo Librandi_, Feb 16 2010
%E A173307 Incorrect formulas and examples deleted by _R. J. Mathar_, Jan 04 2011