A056000 - OEIS
0, 5, 11, 18, 26, 35, 45, 56, 68, 81, 95, 110, 126, 143, 161, 180, 200, 221, 243, 266, 290, 315, 341, 368, 396, 425, 455, 486, 518, 551, 585, 620, 656, 693, 731, 770, 810, 851, 893, 936, 980, 1025, 1071, 1118, 1166, 1215, 1265, 1316, 1368, 1421, 1475
REFERENCES
Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, p. 193.
FORMULA
G.f.: x(5-4x)/(1-x)^3.
(End)
a(n) = binomial(n,2) - 4*n, n >= 9. - Zerinvary Lajos, Nov 25 2006
If we define 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-1,5), for n >= 1. - Milan Janjic, Dec 20 2008
a(n) = Sum_{k=1..n} (k+4). - Gary Detlefs, Aug 10 2010
Sum_{n>=1} 1/a(n) = 7129/11340. - R. J. Mathar, Jul 14 2012
a(n) = 5n - floor(n/2) + floor(n^2/2). - Wesley Ivan Hurt, Jun 15 2013
E.g.f.: (1/2)*(x^2 + 10*x)*exp(x). - G. C. Greubel, Jul 17 2017
Sum_{n>=1} (-1)^(n+1)/a(n) = 4*log(2)/9 - 1879/11340. - Amiram Eldar, Jul 03 2020
Product_{n>=1} (1 - 1/a(n)) = -567*cos(sqrt(89)*Pi/2)/(220*Pi).
Product_{n>=1} (1 + 1/a(n)) = 35*cos(sqrt(73)*Pi/2)/(4*Pi). (End)
MATHEMATICA
Table[n (n + 9)/2, {n, 0, 50}] (* or *)
FoldList[#1 + #2 + 4 &, Range[0, 50]] (* or *)
Table[PolygonalNumber[n + 4] - 10, {n, 0, 50}] (* or *)
CoefficientList[Series[x (5 - 4 x)/(1 - x)^3, {x, 0, 50}], x] (* Michael De Vlieger, Jul 30 2017 *)