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 A139273 #44 Sep 08 2022 08:45:33
%S A139273 0,5,26,63,116,185,270,371,488,621,770,935,1116,1313,1526,1755,2000,
%T A139273 2261,2538,2831,3140,3465,3806,4163,4536,4925,5330,5751,6188,6641,
%U A139273 7110,7595,8096,8613,9146,9695,10260,10841,11438,12051,12680
%N A139273 a(n) = n*(8*n - 3).
%C A139273 Sequence found by reading the line from 0, in the direction 0, 5, ..., in the square spiral whose vertices are the triangular numbers A000217. Opposite numbers to the members of A139277 in the same spiral.
%C A139273 Also, sequence of numbers of the form d*A000217(n-1) + 5*n with generating functions x*(5+(d-5)*x)/(1-x)^3; the inverse binomial transform is 0,5,d,0,0,.. (0 continued). See Crossrefs. - _Bruno Berselli_, Feb 11 2011
%C A139273 Even decagonal numbers divided by 2. - _Omar E. Pol_, Aug 19 2011
%H A139273 G. C. Greubel, <a href="/A139273/b139273.txt">Table of n, a(n) for n = 0..5000</a>
%H A139273 Omar E. Pol, <a href="http://www.polprimos.com">Determinacion geometrica de los numeros primos y perfectos</a>.
%H A139273 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A139273 a(n) = 8*n^2 - 3*n.
%F A139273 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) = 3*a(n-1) - 3*a(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 A139273 a(n) = 16*n + a(n-1) - 11 for n>0, a(0)=0. - _Vincenzo Librandi_, Aug 03 2010
%F A139273 From _Bruno Berselli_, Feb 11 2011: (Start)
%F A139273 G.f.: x*(5 + 11*x)/(1 - x)^3.
%F A139273 a(n) = 4*A000217(n) + A051866(n). (End)
%F A139273 a(n) = A028994(n)/2. - _Omar E. Pol_, Aug 19 2011
%F A139273 a(0)=0, a(1)=5, a(2)=26; for n>2, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - _Harvey P. Dale_, Feb 02 2012
%F A139273 E.g.f.: (8*x^2 + 5*x)*exp(x). - _G. C. Greubel_, Jul 18 2017
%F A139273 Sum_{n>=1} 1/a(n) = 4*log(2)/3 - (sqrt(2)-1)*Pi/6 - sqrt(2)*arccoth(sqrt(2))/3. - _Amiram Eldar_, Jul 03 2020
%t A139273 Table[n (8 n - 3), {n, 0, 40}] (* or *) LinearRecurrence[{3, -3, 1}, {0, 5, 26}, 40] (* _Harvey P. Dale_, Feb 02 2012 *)
%o A139273 (Magma) [ n*(8*n-3) : n in [0..40] ]; // _Bruno Berselli_, Feb 11 2011
%o A139273 (PARI) a(n)=n*(8*n-3) \\ _Charles R Greathouse IV_, Sep 24 2015
%Y A139273 Cf. A000217, A014634, A014635, A033585, A033586, A033587, A035008, A051870, A069129, A085250, A072279, A139272, A139274, A139275, A139276, A139278, A139279, A139280, A139281, A139282.
%Y A139273 Cf. numbers of the form n*(d*n+10-d)/2: A008587, A056000, A028347, A140090, A014106, A028895, A045944, A186029, A007742, A022267, A033429, A022268, A049452, A186030, A135703, A152734.
%K A139273 nonn,easy
%O A139273 0,2
%A A139273 _Omar E. Pol_, Apr 26 2008