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%I #32 Sep 08 2022 08:46:08
%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
%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 Cf. A000290, A001105, A010021, A016742, A016802, A139098.
%Y Pentasection of A077221, A181900.
%K nonn,easy
%O 0,2
%A _Wesley Ivan Hurt_, Jun 19 2014
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