A033569 - OEIS

%I #44 Apr 22 2024 06:34:40

%S -1,4,21,50,91,144,209,286,375,476,589,714,851,1000,1161,1334,1519,

%T 1716,1925,2146,2379,2624,2881,3150,3431,3724,4029,4346,4675,5016,

%U 5369,5734,6111,6500,6901,7314,7739,8176,8625,9086,9559,10044,10541,11050,11571

%N a(n) = (2*n - 1)*(3*n + 1).

%C For n>0, a(n) is the sum of the numbers from 2n+2 to 4n. The last digit of a(n) corresponds to the last digit of the squares mod 10 (A008959). Binomial Transform of a(n) starts: -1, 3, 28, 124, 432, 1328, 3776, 10176, 26368, ... . - _Wesley Ivan Hurt_, Dec 06 2014

%H Vincenzo Librandi, <a href="/A033569/b033569.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.: (-1+7*x+6*x^2)/(1-x)^3. - _Vincenzo Librandi_, Jul 07 2012

%F a(n) = 3*a(n-1) -3*a(n-2) +a(n-3). - _Vincenzo Librandi_, Jul 07 2012

%F E.g.f.: (-1+5*x+6*x^2)*e^x. - _Robert Israel_, Dec 07 2014

%F a(n) = A060747(n) * A016777(n). - _Reinhard Zumkeller_, Jul 05 2015

%F Sum_{n>=0} 1/a(n) = 2/5*(log(2)-1) -sqrt(3)*Pi/30 -3*log(3)/10 = -0.6337047... - _R. J. Mathar_, Apr 22 2024

%p A033569:=n->(2*n-1)*(3*n+1): seq(A033569(n), n=0..50); # _Wesley Ivan Hurt_, Dec 06 2014

%t CoefficientList[Series[(-1+7*x+6*x^2)/(1-x)^3,{x,0,50}],x] (* _Vincenzo Librandi_, Jul 07 2012 *)

%o (Magma) [(2*n-1)*(3*n+1): n in [0..50]]; // _Vincenzo Librandi_, Jul 07 2012

%o (Haskell)

%o a033569 n = (2 * n - 1) * (3 * n + 1)

%o a033569_list = map a033569 [0..]

%o -- _Reinhard Zumkeller_, Jul 05 2015

%o (PARI) a(n)=(2*n-1)*(3*n+1) \\ _Charles R Greathouse IV_, Jun 17 2017

%o (Sage) [(2*n-1)*(3*n+1) for n in (0..50)] # _G. C. Greubel_, Apr 02 2019

%o (GAP) List([0..50], n-> (2*n-1)*(3*n+1)) # _G. C. Greubel_, Apr 02 2019

%Y Cf. A008959, A060747, A016777, A259758 (subsequence).

%K sign,easy

%O 0,2

%A _N. J. A. Sloane_