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 A139272 #43 Sep 27 2024 05:37:51
%S A139272 0,3,22,57,108,175,258,357,472,603,750,913,1092,1287,1498,1725,1968,
%T A139272 2227,2502,2793,3100,3423,3762,4117,4488,4875,5278,5697,6132,6583,
%U A139272 7050,7533,8032,8547,9078,9625,10188,10767,11362,11973,12600
%N A139272 a(n) = n*(8*n-5).
%C A139272 Sequence found by reading the line from 0, in the direction 0, 3, ..., in the square spiral whose vertices are the triangular numbers A000217. Opposite numbers to the members of A139276 in the same spiral.
%H A139272 G. C. Greubel, <a href="/A139272/b139272.txt">Table of n, a(n) for n = 0..5000</a>
%H A139272 Omar E. Pol, <a href="http://www.polprimos.com">Determinacion geometrica de los numeros primos y perfectos</a>.
%H A139272 Franck Ramaharo, <a href="https://arxiv.org/abs/1802.07701">Statistics on some classes of knot shadows</a>, arXiv:1802.07701 [math.CO], 2018.
%H A139272 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A139272 a(n) = 8*n^2 - 5*n.
%F A139272 Sequences of the form a(n) = 8*n^2 + c*n have generating functions x{c+8+(8-c)x} / (1-x)^3 and recurrence a(n) = 3a(n-1) - 3a(n-2) + a(n-3). The inverse binomial transform is 0, c+8, 16, 0, 0, ... (0 continued). This applies to A139271-A139278, positive or negative c. - _R. J. Mathar_, May 12 2008
%F A139272 a(n) = 16*n + a(n-1) - 13 with n>0, a(0)=0. - _Vincenzo Librandi_, Aug 03 2010
%F A139272 From _G. C. Greubel_, Jul 18 2017: (Start)
%F A139272 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
%F A139272 G.f.: x*(13*x + 3)/(1-x)^3.
%F A139272 E.g.f.: (8*x^2 + 3*x)*exp(x). (End)
%F A139272 Sum_{n>=1} 1/a(n) = ((sqrt(2)-1)*Pi + 8*log(2) - 2*sqrt(2)*log(sqrt(2)+1))/10. - _Amiram Eldar_, Mar 17 2022
%t A139272 s=0;lst={s};Do[s+=n++ +3;AppendTo[lst, s], {n, 0, 7!, 16}];lst (* _Vladimir Joseph Stephan Orlovsky_, Nov 16 2008 *)
%t A139272 Table[n*(8*n -5), {n,0,50}] (* _G. C. Greubel_, Jul 18 2017 *)
%t A139272 LinearRecurrence[{3,-3,1},{0,3,22},50] (* _Harvey P. Dale_, Jan 13 2024 *)
%o A139272 (PARI) a(n)=n*(8*n-5) \\ _Charles R Greathouse IV_, Oct 07 2015
%Y A139272 Cf. A000217, A014634, A014635, A033585, A033586, A033587, A035008, A051870, A069129, A085250, A072279, A139273, A139274, A139275, A139276, A139278, A139279, A139280, A139281, A139282.
%Y A139272 Cf. numbers of the form n*(n*k-k+6)/2, this sequence is the case k=16: see Comments lines of A226492.
%K A139272 nonn,easy
%O A139272 0,2
%A A139272 _Omar E. Pol_, Apr 26 2008