|
%I #38 Jul 30 2024 04:28:07
%S 0,12,30,55,88,130,182,245,320,408,510,627,760,910,1078,1265,1472,
%T 1700,1950,2223,2520,2842,3190,3565,3968,4400,4862,5355,5880,6438,
%U 7030,7657,8320,9020,9758,10535,11352,12210,13110,14053,15040,16072,17150,18275,19448
%N a(n) = n*(n+7)*(n+8)/6.
%H Vincenzo Librandi, <a href="/A111396/b111396.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F a(n) = binomial(n+8,3) - 2*binomial(n+8,2). - _Zerinvary Lajos_, Nov 25 2006, corrected by _R. J. Mathar_, Mar 15 2011
%F G.f.: x*(12 - 18*x + 7*x^2) /(x-1)^4. - _R. J. Mathar_, Mar 15 2011
%F a(n) = 4*a(n-1) -6*a(n-2) +4*a(n-3) -a(n-4). - _Vincenzo Librandi_, Jun 27 2012
%F E.g.f.: (x/6)*(72 + 18*x + x^2)*exp(x). - _G. C. Greubel_, Jul 30 2022
%F From _Amiram Eldar_, Jul 30 2024: (Start)
%F Sum_{n>=1} 1/a(n) = 1443/7840.
%F Sum_{n>=1} (-1)^(n+1)/a(n) = 12*log(2)/7 - 1767/1568. (End)
%t Table[n(n+7)(n+8)/6, {n,0,100}] (* _Vladimir Joseph Stephan Orlovsky_, Jul 06 2011 *)
%t CoefficientList[Series[x*(12-18*x+7*x^2)/(x-1)^4,{x,0,50}],x] (* or *) LinearRecurrence[{4,-6,4,-1},{0,12,30,55},40] (* _Vincenzo Librandi_, Jun 27 2012 *)
%o (Magma) I:=[0, 12, 30, 55]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..50]]; // _Vincenzo Librandi_, Jun 27 2012
%o (PARI) a(n)=n*(n+7)*(n+8)/6 \\ _Charles R Greathouse IV_, Oct 16 2015
%o (SageMath) [n*(n+7)*(n+8)/6 for n in (0..50)] # _G. C. Greubel_, Jul 30 2022
%Y Cf. A111373.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_, Nov 11 2005
|
