A195824 - OEIS

%I #31 Dec 01 2024 14:52:00

%S 0,24,96,216,384,600,864,1176,1536,1944,2400,2904,3456,4056,4704,5400,

%T 6144,6936,7776,8664,9600,10584,11616,12696,13824,15000,16224,17496,

%U 18816,20184,21600,23064,24576,26136,27744,29400,31104,32856,34656,36504,38400,40344

%N a(n) = 24*n^2.

%C Sequence found by reading the line from 0, in the direction 0, 24, ..., in the square spiral whose vertices are the generalized tetradecagonal numbers A195818.

%C Surface area of a cube with side 2n. - _Wesley Ivan Hurt_, Aug 05 2014

%H Vincenzo Librandi, <a href="/A195824/b195824.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*A000290(n) = 12*A001105(n) = 8*A033428(n) = 6*A016742(n) = 4*A033581(n) = 3*A139098(n) = 2*A135453(n).

%F From _Wesley Ivan Hurt_, Aug 05 2014: (Start)

%F G.f.: 24*x*(1+x)/(1-x)^3.

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

%F From _Elmo R. Oliveira_, Dec 01 2024: (Start)

%F E.g.f.: 24*x*(1 + x)*exp(x).

%F a(n) = n*A008606(n) = A195158(2*n). (End)

%p A195824:=n->24*n^2: seq(A195824(n), n=0..50); # _Wesley Ivan Hurt_, Aug 05 2014

%t 24 Range[0, 30]^2 (* or *) Table[24 n^2, {n, 0, 30}] (* or *) CoefficientList[Series[24 x (1 + x)/(1 - x)^3, {x, 0, 30}], x] (* _Wesley Ivan Hurt_, Aug 05 2014 *)

%t LinearRecurrence[{3,-3,1},{0,24,96},40] (* _Harvey P. Dale_, Nov 11 2017 *)

%o (Magma) [24*n^2 : n in [0..50]]; // _Wesley Ivan Hurt_, Aug 05 2014

%o (PARI) a(n) = 24*n^2; \\ _Michel Marcus_, Aug 05 2014

%o (Magma) I:=[0,24,96]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..50]]; // _Vincenzo Librandi_, Aug 06 2014

%Y Bisection of A195158.

%Y Cf. A000290, A001105, A008606, A016742, A033428, A033581, A135453, A139098, A195818.

%K nonn,easy

%O 0,2

%A _Omar E. Pol_, Sep 28 2011