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 A139277 #39 Dec 24 2024 22:12:35
%S A139277 0,13,42,87,148,225,318,427,552,693,850,1023,1212,1417,1638,1875,2128,
%T A139277 2397,2682,2983,3300,3633,3982,4347,4728,5125,5538,5967,6412,6873,
%U A139277 7350,7843,8352,8877,9418,9975,10548,11137,11742,12363,13000
%N A139277 a(n) = n*(8*n+5).
%C A139277 Sequence found by reading the line from 0, in the direction 0, 13, ..., in the square spiral whose vertices are the triangular numbers A000217. Opposite numbers to the members of A139273 in the same spiral.
%H A139277 Omar E. Pol, <a href="http://www.polprimos.com">Determinacion geometrica de los numeros primos y perfectos</a>.
%H A139277 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A139277 a(n) = 8*n^2 + 5*n.
%F A139277 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 A139277 a(n) = 16*n + a(n-1) - 3 for n > 0, a(0)=0. - _Vincenzo Librandi_, Aug 03 2010
%F A139277 Sum_{n>=1} 1/a(n) = (sqrt(2)-1)*Pi/10 - 4*log(2)/5 + sqrt(2)*log(sqrt(2)+1)/5 + 8/25. - _Amiram Eldar_, Mar 18 2022
%F A139277 E.g.f.: exp(x)*x*(13 + 8*x). - _Elmo R. Oliveira_, Dec 15 2024
%t A139277 Table[n (8 n + 5), {n, 0, 50}] (* _Bruno Berselli_, Aug 22 2018 *)
%t A139277 LinearRecurrence[{3,-3,1},{0,13,42},50] (* _Harvey P. Dale_, Dec 04 2018 *)
%o A139277 (PARI) a(n)=n*(8*n+5) \\ _Charles R Greathouse IV_, Jun 17 2017
%Y A139277 Cf. A000217, A014634, A014635, A033585, A033586, A033587, A035008, A051870, A069129, A085250.
%Y A139277 Cf. A139271, A139272, A139273, A139274, A139275, A139276, A139278, A139279, A139280, A139281, A139282.
%K A139277 nonn,easy
%O A139277 0,2
%A A139277 _Omar E. Pol_, Apr 26 2008