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 A139271 #55 Aug 14 2025 20:42:04
%S A139271 0,2,20,54,104,170,252,350,464,594,740,902,1080,1274,1484,1710,1952,
%T A139271 2210,2484,2774,3080,3402,3740,4094,4464,4850,5252,5670,6104,6554,
%U A139271 7020,7502,8000,8514,9044,9590,10152,10730,11324,11934,12560,13202,13860,14534,15224
%N A139271 a(n) = 2*n*(4*n-3).
%C A139271 Sequence found by reading the line from 0, in the direction 0, 2, ..., in the square spiral whose vertices are the triangular numbers A000217. Opposite numbers to the members of A033585 in the same spiral.
%C A139271 Twice decagonal numbers (or twice 10-gonal numbers). - _Omar E. Pol_, May 15 2008
%C A139271 a(n) is the number of walks in a cubic lattice of n dimensions that reach the point of origin for the first time after 4 steps. - _Shel Kaphan_, Mar 20 2023
%H A139271 G. C. Greubel, <a href="/A139271/b139271.txt">Table of n, a(n) for n = 0..5000</a>
%H A139271 Omar E. Pol, <a href="http://www.polprimos.com">Determinacion geometrica de los numeros primos y perfectos</a>.
%H A139271 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A139271 a(n) = 8*n^2 - 6*n.
%F A139271 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 A139271 a(n) = A001107(n)*2. - _Omar E. Pol_, May 15 2008
%F A139271 a(n) = 16*n + a(n-1) - 14 (with a(0)=0). - _Vincenzo Librandi_, Aug 03 2010
%F A139271 From _G. C. Greubel_, Jul 18 2017: (Start)
%F A139271 G.f.: (2*x)*(7*x+1)/(1-x)^3.
%F A139271 E.g.f.: (8*x^2 + 2*x)*exp(x). (End)
%F A139271 Sum_{n>=1} 1/a(n) = Pi/12 + log(2)/2. - _Amiram Eldar_, Mar 28 2023
%t A139271 Table[8n^2-6n,{n,0,40}] (* or *) LinearRecurrence[{3,-3,1},{0,2,20},50] (* _Harvey P. Dale_, Sep 26 2016 *)
%o A139271 (PARI) a(n)=2*n*(4*n-3) \\ _Charles R Greathouse IV_, Jun 17 2017
%Y A139271 Cf. A000217, A014634, A014635, A033585, A033586, A033587, A035008, A051870, A069129, A085250, A139272, A139273, A139274, A139275, A139276, A139278, A139279, A139280, A139281, A139282.
%Y A139271 Cf. A001107.
%Y A139271 Cf. numbers of the form n*(n*k-k+4)/2 listed in A226488 (this sequence is the case k=16). - _Bruno Berselli_, Jun 10 2013
%Y A139271 Row n=2 of A361397.
%K A139271 easy,nonn,walk
%O A139271 0,2
%A A139271 _Omar E. Pol_, Apr 26 2008
%E A139271 Corrected by _Harvey P. Dale_, Sep 26 2016