A139275 - OEIS

%I #59 Apr 15 2022 01:43:40

%S 0,9,34,75,132,205,294,399,520,657,810,979,1164,1365,1582,1815,2064,

%T 2329,2610,2907,3220,3549,3894,4255,4632,5025,5434,5859,6300,6757,

%U 7230,7719,8224,8745,9282,9835,10404,10989,11590,12207,12840

%N a(n) = n*(8*n+1).

%C Sequence found by reading the line from 0, in the direction 0, 9,..., in the square spiral whose vertices are the triangular numbers A000217.

%H G. C. Greubel, <a href="/A139275/b139275.txt">Table of n, a(n) for n = 0..5000</a>

%H Omar E. Pol, <a href="http://www.polprimos.com">Determinacion geometrica de los numeros primos y perfectos</a>.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

%F a(n) = 8*n^2 + n.

%F 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 a(n) = 16*n + a(n-1) - 7 with n>0, a(0)=0. - _Vincenzo Librandi_, Aug 03 2010

%F a(n) = A000217(5*n) - A000217(3*n). - _Bruno Berselli_, Sep 21 2016

%F Sum_{n>=1} 1/a(n) = 8 - (1+sqrt(2))*Pi/2 - 4*log(2) - sqrt(2) * log(1+sqrt(2)) = 0.1887230016056779928... . - _Vaclav Kotesovec_, Sep 21 2016

%F From _G. C. Greubel_, Jul 18 2017: (Start)

%F G.f.: x*(7*x + 9)/(1-x)^3.

%F E.g.f.: (8*x^2 + 9*x)*exp(x). (End)

%t Table[n (8 n + 1), {n, 0, 40}] (* _Bruno Berselli_, Sep 21 2016 *)

%t LinearRecurrence[{3,-3,1},{0,9,34},50] (* _Harvey P. Dale_, Apr 21 2020 *)

%o (PARI) a(n) = n*(8*n+1); \\ _Altug Alkan_, Sep 21 2016

%Y Cf. A000217, A014634, A014635, A033585, A033586, A033587, A035008, A051870, A069129, A085250, A072279, A139272, A139273, A139274, A139276, A139278, A139279, A139280, A139281, A139282.

%K nonn,easy

%O 0,2

%A _Omar E. Pol_, Apr 26 2008