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A152741 13 times triangular numbers.

%I A152741 #31 Feb 21 2023 02:12:39
%S A152741 0,13,39,78,130,195,273,364,468,585,715,858,1014,1183,1365,1560,1768,
%T A152741 1989,2223,2470,2730,3003,3289,3588,3900,4225,4563,4914,5278,5655,
%U A152741 6045,6448,6864,7293,7735,8190,8658,9139,9633,10140,10660,11193,11739,12298,12870
%N A152741 13 times triangular numbers.
%C A152741 Sequence found by reading the line from 0, in the direction 0, 13,... and the same line from 0, in the direction 0, 39,..., in the square spiral whose vertices are the generalized 15-gonal numbers. - _Omar E. Pol_, Oct 03 2011
%C A152741 Sum of the numbers from 6n to 7n. - _Wesley Ivan Hurt_, Dec 22 2015
%H A152741 G. C. Greubel, <a href="/A152741/b152741.txt">Table of n, a(n) for n = 0..5000</a>
%H A152741 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A152741 a(n) = 13*n*(n+1)/2 = 13 * A000217(n).
%F A152741 a(n) = a(n-1)+13*n (with a(0)=0). - _Vincenzo Librandi_, Nov 26 2010
%F A152741 a(n) = A069126(n+1) - 1. - _Omar E. Pol_, Oct 03 2011
%F A152741 From _Wesley Ivan Hurt_, Dec 22 2015: (Start)
%F A152741 G.f.: 13*x/(1-x)^3.
%F A152741 a(n) = 3*a(n-1)-3*a(n-2)+a(n-3) for n>2.
%F A152741 a(n) = Sum_{i=6n..7n} i. (End)
%F A152741 E.g.f.: 13*x*(2+x)*exp(x)/2. - _G. C. Greubel_, Sep 01 2018
%F A152741 From _Amiram Eldar_, Feb 21 2023: (Start)
%F A152741 Sum_{n>=1} 1/a(n) = 2/13.
%F A152741 Sum_{n>=1} (-1)^(n+1)/a(n) = (4*log(2) - 2)/13.
%F A152741 Product_{n>=1} (1 - 1/a(n)) = -(13/(2*Pi))*cos(sqrt(21/13)*Pi/2).
%F A152741 Product_{n>=1} (1 + 1/a(n)) = (13/(2*Pi))*cos(sqrt(5/13)*Pi/2). (End)
%p A152741 A152741:=n->13*n*(n+1)/2: seq(A152741(n), n=0..60); # _Wesley Ivan Hurt_, Dec 22 2015
%t A152741 Table[13*n*(n-1)/2, {n,100}] (* _Vladimir Joseph Stephan Orlovsky_, Jul 06 2011 *)
%t A152741 CoefficientList[Series[13 x/(1 - x)^3, {x, 0, 50}], x] (* _Wesley Ivan Hurt_, Dec 22 2015 *)
%o A152741 (Magma) [13*n*(n+1)/2 : n in [0..60]]; // _Wesley Ivan Hurt_, Dec 22 2015
%o A152741 (PARI) a(n)=13*n*(n+1)/2 \\ _Charles R Greathouse IV_, Jun 17 2017
%Y A152741 Cf. A000217, A049598, A069126.
%K A152741 easy,nonn
%O A152741 0,2
%A A152741 _Omar E. Pol_, Dec 12 2008