A244082 - OEIS

%I #35 Dec 02 2024 12:21:43

%S 0,32,128,288,512,800,1152,1568,2048,2592,3200,3872,4608,5408,6272,

%T 7200,8192,9248,10368,11552,12800,14112,15488,16928,18432,20000,21632,

%U 23328,25088,26912,28800,30752,32768,34848,36992,39200,41472,43808,46208,48672,51200

%N a(n) = 32*n^2.

%C Geometric connections of a(n) to the area and perimeter of a square.

%C Area:

%C . half the area of a square with side 8n (cf. A008590);

%C . area of a square with diagonal 8n (cf. A008590);

%C . twice the area of a square with side 4n (cf. A008586);

%C . four times the area of a square with diagonal 4n (cf. A008586);

%C . eight times the area of a square with side 2n (cf. A005843);

%C . sixteen times the area of a square with diagonal 2n (cf. A005843);

%C . thirty two times the area of a square with side n (cf. A001477);

%C . sixty four times the area of a square with diagonal n (cf. A001477).

%C Perimeter:

%C . perimeter of a square with side 8n^2 (cf. A139098);

%C . twice the perimeter of a square with side 4n^2 (cf. A016742);

%C . four times the perimeter of a square with side 2n^2 (cf. A001105);

%C . eight times the perimeter of a square with side n^2 (cf. A000290).

%C Sequence found by reading the line from 0, in the direction 0, 32, ..., in the square spiral whose vertices are the generalized 18-gonal numbers. - _Omar E. Pol_, May 10 2018

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

%F G.f.: 32*x*(1+x)/(1-x)^3.

%F a(n) =  2 * A016802(n).

%F a(n) =  4 * A139098(n).

%F a(n) =  8 * A016742(n).

%F a(n) = 16 * A001105(n).

%F a(n) = 32 * A000290(n).

%F a(n) = A010021(n) - 2 for n > 0. - _Bruno Berselli_, Jun 24 2014

%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - _Wesley Ivan Hurt_, Nov 19 2021

%F From _Elmo R. Oliveira_, Dec 02 2024: (Start)

%F E.g.f.: 32*x*(1 + x)*exp(x).

%F a(n) = n*A174312(n) = A139098(2*n). (End)

%p A244082:=n->32*n^2; seq(A244082(n), n=0..50);

%t 32 Range[0, 50]^2 (* or *)

%t Table[32 n^2, {n, 0, 50}] (* or *)

%t CoefficientList[Series[32 x (1 + x)/(1 - x)^3, {x, 0, 30}], x]

%o (Magma) [32*n^2 : n in [0..50]];

%o (PARI) a(n)=32*n^2 \\ _Charles R Greathouse IV_, Jun 17 2017

%Y Pentasection of A077221, A181900.

%Y Cf. A000290, A001105, A010021, A016742, A016802, A139098.

%Y Cf. A001477, A005843, A008586, A008590, A139098, A174312.

%K nonn,easy

%O 0,2

%A _Wesley Ivan Hurt_, Jun 19 2014