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 A193872 #17 Dec 15 2024 17:15:48 %S A193872 0,12,64,156,288,460,672,924,1216,1548,1920,2332,2784,3276,3808,4380, %T A193872 4992,5644,6336,7068,7840,8652,9504,10396,11328,12300,13312,14364, %U A193872 15456,16588,17760,18972,20224,21516,22848,24220,25632,27084,28576,30108,31680,33292,34944 %N A193872 Even dodecagonal numbers: a(n) = 4*n*(5*n - 2). %C A193872 Even 12-gonal numbers. Bisection of A051624. %H A193872 Harvey P. Dale, <a href="/A193872/b193872.txt">Table of n, a(n) for n = 0..1000</a> %H A193872 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1). %F A193872 a(n) = 4*A147874(n+1). %F A193872 a(n) = 4*n*A016885(n-1), n >= 1. %F A193872 From _Elmo R. Oliveira_, Dec 15 2024: (Start) %F A193872 G.f.: 4*x*(3 + 7*x)/(1 - x)^3. %F A193872 E.g.f.: 4*x*(3 + 5*x)*exp(x). %F A193872 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n >= 3. (End) %t A193872 PolygonalNumber[12,2*Range[0,40]] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Jan 02 2017 *) %o A193872 (PARI) a(n)=4*n*(5*n-2) \\ _Charles R Greathouse IV_, Jun 17 2017 %Y A193872 Cf. A016885, A051624, A147874. %K A193872 nonn,easy %O A193872 0,2 %A A193872 _Omar E. Pol_, Aug 19 2011