%I #34 Apr 10 2022 14:02:41
%S 0,14,29,45,62,80,99,119,140,162,185,209,234,260,287,315,344,374,405,
%T 437,470,504,539,575,612,650,689,729,770,812,855,899,944,990,1037,
%U 1085,1134,1184,1235,1287,1340,1394,1449,1505,1562,1620
%N a(n) = n*(n + 27)/2.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F Let f(n,i,a) = Sum_{k=0..n-i} (binomial(n,k)*Stirling1(n-k,i)*Product_{j=0..k-1} (-a-j)), then a(n) = -f(n,n-1,14), for n>=1. - _Milan Janjic_, Dec 20 2008
%F a(n) = n + a(n-1) + 13 (with a(0)=0). - _Vincenzo Librandi_, Aug 03 2010
%F a(n) = 14*n - floor(n/2) + floor(n^2/2). - _Wesley Ivan Hurt_, Jun 15 2013
%F From _Chai Wah Wu_, Jun 02 2016: (Start)
%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2.
%F G.f.: x*(13*x - 14)/(x - 1)^3. (End)
%F From _Amiram Eldar_, Jan 10 2021: (Start)
%F Sum_{n>=1} 1/a(n) = 2*A001008(27)/(27*A002805(27)) = 312536252003/1084231348200.
%F Sum_{n>=1} (-1)^(n+1)/a(n) = 4*log(2)/27 - 57128792093/1084231348200. (End)
%t Table[(n(n+27))/2,{n,0,50}] (* or *) LinearRecurrence[{3,-3,1},{0,14,29},50] (* _Harvey P. Dale_, Apr 10 2022 *)
%o (PARI) a(n)=n*(n+27)/2 \\ _Charles R Greathouse IV_, Jun 17 2017
%Y Cf. A000217, A001008, A002805, A056126.
%K nonn,easy
%O 0,2
%A _Omar E. Pol_, Aug 28 2007