This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A135453 #68 Dec 01 2024 20:17:34
%S A135453 0,12,48,108,192,300,432,588,768,972,1200,1452,1728,2028,2352,2700,
%T A135453 3072,3468,3888,4332,4800,5292,5808,6348,6912,7500,8112,8748,9408,
%U A135453 10092,10800,11532,12288,13068,13872,14700,15552,16428,17328,18252,19200,20172,21168,22188
%N A135453 a(n) = 12*n^2.
%C A135453 Areas of perfect 4:3 rectangles (for n > 0).
%C A135453 Sequence found by reading the line from 0, in the direction 0, 12, ..., in the square spiral whose vertices are the generalized octagonal numbers A001082. Semi-axis opposite to A069190 in the same spiral. - _Omar E. Pol_, Sep 16 2011
%C A135453 (x,y,z) = (-a(n), 1 + n*a(n), 1 - n*a(n)) are solutions of the Diophantine equation x^3 + 2*y^3 + 2*z^3 = 4. - _XU Pingya_, Apr 30 2022
%H A135453 Ivan Panchenko, <a href="/A135453/b135453.txt">Table of n, a(n) for n = 0..10000</a>
%H A135453 John Elias, <a href="/A135453/a135453.png">Illustration: Even Ordered Star Perimeters</a>.
%H A135453 Leo Tavares, <a href="/A135453/a135453.jpg">Illustration</a>.
%H A135453 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A135453 a(n) = 12*A000290(n) = 6*A001105(n) = 4*A033428(n) = 3*A016742(n) = 2*A033581(n). - _Omar E. Pol_, Dec 13 2008
%F A135453 From _Amiram Eldar_, Feb 03 2021: (Start)
%F A135453 Sum_{n>=1} 1/a(n) = Pi^2/72 (A086729).
%F A135453 Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/144.
%F A135453 Product_{n>=1} (1 + 1/a(n)) = 2*sqrt(3)*sinh(Pi/(2*sqrt(3)))/Pi.
%F A135453 Product_{n>=1} (1 - 1/a(n)) = 2*sqrt(3)*sin(Pi/(2*sqrt(3)))/Pi. (End)
%F A135453 From _Elmo R. Oliveira_, Nov 30 2024: (Start)
%F A135453 G.f.: 12*x*(1 + x)/(1-x)^3.
%F A135453 E.g.f.: 12*x*(1 + x)*exp(x).
%F A135453 a(n) = n*A008594(n) = A195143(2*n).
%F A135453 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 2. (End)
%e A135453 192 is on the list since 16*12 is a 4:3 rectangle with integer sides and an area of 192.
%p A135453 seq(12*h^2,n=0..100); # _Muniru A Asiru_, Jan 29 2018
%t A135453 Table[12*n^2, {n, 0, 60}] (* _Stefan Steinerberger_, Dec 17 2007 *)
%t A135453 LinearRecurrence[{3,-3,1},{0,12,48},50] (* _Harvey P. Dale_, Jan 19 2020 *)
%o A135453 (PARI) a(n)=12*n^2 \\ _Charles R Greathouse IV_, Jun 17 2017
%o A135453 (GAP) List([0..100],n->12*n^2); # _Muniru A Asiru_, Jan 29 2018
%Y A135453 Cf. A000290, A001105, A016742, A033428, A033581, A069190, A086729.
%Y A135453 Cf. A008594, A195143.
%K A135453 nonn,easy
%O A135453 0,2
%A A135453 _Ben Paul Thurston_, Dec 14 2007
%E A135453 More terms from _Stefan Steinerberger_, Dec 17 2007
%E A135453 Minor edits from _Omar E. Pol_, Dec 15 2008