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%I #47 Sep 08 2022 08:45:01
%S 0,9,19,30,42,55,69,84,100,117,135,154,174,195,217,240,264,289,315,
%T 342,370,399,429,460,492,525,559,594,630,667,705,744,784,825,867,910,
%U 954,999,1045,1092,1140,1189,1239,1290,1342,1395,1449,1504,1560,1617,1675
%N a(n) = n*(n + 17)/2.
%H G. C. Greubel, <a href="/A056126/b056126.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F G.f.: x*(9-8*x)/(1-x)^3.
%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
%F a(n) = A126890(n,8) for n>7. - _Reinhard Zumkeller_, Dec 30 2006
%F 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-1,9), for n>=1. - _Milan Janjic_, Dec 20 2008
%F a(n) = a(n-1) + n + 8 (with a(0)=0). - _Vincenzo Librandi_, Aug 07 2010
%F a(n) = 9*n - floor(n/2) + floor(n^2/2). - _Wesley Ivan Hurt_, Jun 15 2013
%F E.g.f.: x*(18 + x)*exp(x)/2. - _G. C. Greubel_, Jan 19 2020
%F From _Amiram Eldar_, Jan 10 2021: (Start)
%F Sum_{n>=1} 1/a(n) = 2*A001008(17)/(17*A002805(17)) = 42142223/104144040.
%F Sum_{n>=1} (-1)^(n+1)/a(n) = 4*log(2)/17 - 1768477/20828808. (End)
%p seq( n*(n+17)/2, n=0..50); # _G. C. Greubel_, Jan 19 2020
%t Table[n(n+17)/2,{n,0,50}] (* _Harvey P. Dale_, Apr 25 2011 *)
%o (PARI) a(n)=n*(n+17)/2 \\ _Charles R Greathouse IV_, Sep 24 2015
%o (Magma) [n*(n+17)/2: n in [0..50]]; // _G. C. Greubel_, Jan 19 2020
%o (Sage) [n*(n+17)/2 for n in (0..50)] # _G. C. Greubel_, Jan 19 2020
%o (GAP) List([0..50], n-> n*(n+17)/2 ); # _G. C. Greubel_, Jan 19 2020
%Y Cf. A000096, A001477, A051942, A056000, A056121.
%Y Cf. A001008, A002805, A098849, A120071, A132760, A132761, A132765.
%K easy,nonn
%O 0,2
%A _Barry E. Williams_, Jul 07 2000
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