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 A139276 #36 Jul 02 2022 11:36:59
%S A139276 0,11,38,81,140,215,306,413,536,675,830,1001,1188,1391,1610,1845,2096,
%T A139276 2363,2646,2945,3260,3591,3938,4301,4680,5075,5486,5913,6356,6815,
%U A139276 7290,7781,8288,8811,9350,9905,10476,11063,11666,12285,12920
%N A139276 a(n) = n*(8*n+3).
%C A139276 Sequence found by reading the line from 0, in the direction 0, 11,..., in the square spiral whose vertices are the triangular numbers A000217. Opposite numbers to the members of A139272 in the same spiral.
%H A139276 G. C. Greubel, <a href="/A139276/b139276.txt">Table of n, a(n) for n = 0..5000</a>
%H A139276 Omar E. Pol, <a href="http://www.polprimos.com">Determinacion geometrica de los numeros primos y perfectos</a>.
%H A139276 Amelia Carolina Sparavigna, <a href="https://doi.org/10.5281/zenodo.3470205">The groupoid of the Triangular Numbers and the generation of related integer sequences</a>, Politecnico di Torino, Italy (2019).
%H A139276 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A139276 a(n) = 8*n^2 + 3*n.
%F A139276 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 A139276 a(n) = 16*n+a(n-1)-5 (with a(0)=0). - _Vincenzo Librandi_, Aug 03 2010
%F A139276 From _G. C. Greubel_, Jul 18 2017: (Start)
%F A139276 G.f.: x*(5*x + 11)/(1-x)^3.
%F A139276 E.g.f.: (8*x^2 + 11*x)*exp(x). (End)
%F A139276 Sum_{n>=1} 1/a(n) = 8/9 - (sqrt(2)-1)*Pi/6 - 4*log(2)/3 + sqrt(2)*log(sqrt(2)+1)/3. - _Amiram Eldar_, Mar 17 2022
%e A139276 a(1)=16*1+0-5=11; a(2)=16*2+11-5=38; a(3)=16*3+38-5=81. - _Vincenzo Librandi_, Aug 03 2010
%t A139276 a[n_]:=n*(8*n+3); a[Range[0,60]] (* _Vladimir Joseph Stephan Orlovsky_, Feb 05 2011*)
%t A139276 LinearRecurrence[{3,-3,1},{0,11,38},50] (* _Harvey P. Dale_, Jul 02 2022 *)
%o A139276 (PARI) a(n)=n*(8*n+3) \\ _Charles R Greathouse IV_, Jun 16 2017
%Y A139276 Cf. A000217, A014634, A014635, A033585, A033586, A033587, A035008, A051870, A069129, A085250, A072279, A139272, A139273, A139274, A139275, A139277, A139278, A139279, A139280, A139281, A139282.
%K A139276 easy,nonn
%O A139276 0,2
%A A139276 _Omar E. Pol_, Apr 26 2008