|
%I #21 May 31 2021 03:56:39
%S 0,12,72,180,336,540,792,1092,1440,1836,2280,2772,3312,3900,4536,5220,
%T 5952,6732,7560,8436,9360,10332,11352,12420,13536,14700,15912,17172,
%U 18480,19836,21240,22692,24192,25740,27336,28980,30672,32412
%N 12 times hexagonal numbers: 12*n*(2*n-1).
%C Sequence found by reading the line from 0, in the direction 0, 12,..., in the square spiral whose vertices are the generalized tetradecagonal numbers A195818. - _Omar E. Pol_, Oct 02 2011
%H Ivan Panchenko, <a href="/A143698/b143698.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 a(n) = 24*n^2 - 12*n = 12*A000384(n) = 6*A002939(n) = 4*A094159(n) = 3*A085250(n) = 2*A152746(n).
%F a(n) = a(n-1) + 48*n - 36, with a(0)=0. - _Vincenzo Librandi_, Dec 14 2010
%F From _G. C. Greubel_, May 30 2021: (Start)
%F G.f.: 12*x*(1 + 3*x)/(1-x)^3.
%F E.g.f.: 12*x*(1 + 2*x)*exp(x). (End)
%p seq(12*n*(2*n-1), n=0..40); # _G. C. Greubel_, May 30 2021
%t Table[24n^2-12n,{n,0,40}] (* or *) LinearRecurrence[{3,-3,1},{0,12,72},40] (* _Harvey P. Dale_, Sep 24 2015 *)
%o (PARI) a(n)=24*n^2-12*n \\ _Charles R Greathouse IV_, Jun 17 2017
%o (Sage) [12*n*(2*n-1) for n in (0..40)] # _G. C. Greubel_, May 30 2021
%Y Cf. A000384, A002939, A085250, A094159, A152746, A154617.
%K easy,nonn
%O 0,2
%A _Omar E. Pol_, Jan 23 2009
|
