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