A132756 - OEIS

%I #37 Jan 12 2025 21:18:29

%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)

%F From _Elmo R. Oliveira_, Jan 12 2025: (Start)

%F E.g.f.: exp(x)*x*(28 + x)/2.

%F a(n) = A132769(n)/2. (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, A132769.

%K nonn,easy,changed

%O 0,2

%A _Omar E. Pol_, Aug 28 2007