A139275 a(n) = n*(8*n+1).

0, 9, 34, 75, 132, 205, 294, 399, 520, 657, 810, 979, 1164, 1365, 1582, 1815, 2064, 2329, 2610, 2907, 3220, 3549, 3894, 4255, 4632, 5025, 5434, 5859, 6300, 6757, 7230, 7719, 8224, 8745, 9282, 9835, 10404, 10989, 11590, 12207, 12840

Offset

0,2

Comments

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

Links

G. C. Greubel, Table of n, a(n) for n = 0..5000

Omar E. Pol, Determinacion geometrica de los numeros primos y perfectos.

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Formula

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

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

a(n) = 16*n + a(n-1) - 7 with n>0, a(0)=0. - Vincenzo Librandi, Aug 03 2010

a(n) = A000217(5*n) - A000217(3*n). - Bruno Berselli, Sep 21 2016

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

From G. C. Greubel, Jul 18 2017: (Start)

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

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

Mathematica

Table[n (8 n + 1), {n, 0, 40}] (* Bruno Berselli, Sep 21 2016 *)
LinearRecurrence[{3, -3, 1}, {0, 9, 34}, 50] (* Harvey P. Dale, Apr 21 2020 *)

Prog

(PARI) a(n) = n*(8*n+1); \\ Altug Alkan, Sep 21 2016

Crossrefs

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

Sequence in context: A044467 A020163 A262959 * A236370 A273744 A133547

Adjacent sequences: A139272 A139273 A139274 * A139276 A139277 A139278

Keyword

nonn,easy

Author

Omar E. Pol, Apr 26 2008

Status

approved