A139272 a(n) = n*(8*n-5).

0, 3, 22, 57, 108, 175, 258, 357, 472, 603, 750, 913, 1092, 1287, 1498, 1725, 1968, 2227, 2502, 2793, 3100, 3423, 3762, 4117, 4488, 4875, 5278, 5697, 6132, 6583, 7050, 7533, 8032, 8547, 9078, 9625, 10188, 10767, 11362, 11973, 12600

Offset

0,2

Comments

Sequence found by reading the line from 0, in the direction 0, 3, ..., in the square spiral whose vertices are the triangular numbers A000217. Opposite numbers to the members of A139276 in the same spiral.

Links

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

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

Franck Ramaharo, Statistics on some classes of knot shadows, arXiv:1802.07701 [math.CO], 2018.

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

Formula

a(n) = 8*n^2 - 5*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) = 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

a(n) = 16*n + a(n-1) - 13 with n>0, a(0)=0. - Vincenzo Librandi, Aug 03 2010

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

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).

G.f.: x*(13*x + 3)/(1-x)^3.

E.g.f.: (8*x^2 + 3*x)*exp(x). (End)

Sum_{n>=1} 1/a(n) = ((sqrt(2)-1)*Pi + 8*log(2) - 2*sqrt(2)*log(sqrt(2)+1))/10. - Amiram Eldar, Mar 17 2022

Mathematica

s=0; lst={s}; Do[s+=n++ +3; AppendTo[lst, s], {n, 0, 7!, 16}]; lst (* Vladimir Joseph Stephan Orlovsky, Nov 16 2008 *)
Table[n*(8*n -5), {n, 0, 50}] (* G. C. Greubel, Jul 18 2017 *)
LinearRecurrence[{3, -3, 1}, {0, 3, 22}, 50] (* Harvey P. Dale, Jan 13 2024 *)

Prog

(PARI) a(n)=n*(8*n-5) \\ Charles R Greathouse IV, Oct 07 2015

Crossrefs

Cf. A000217, A014634, A014635, A033585, A033586, A033587, A035008, A051870, A069129, A085250, A072279, A139273, A139274, A139275, A139276, A139278, A139279, A139280, A139281, A139282.

Cf. numbers of the form n*(n*k-k+6))/2, this sequence is the case k=16: see Comments lines of A226492.

Sequence in context: A225414 A187694 A104604 * A159345 A006532 A274870

Adjacent sequences: A139269 A139270 A139271 * A139273 A139274 A139275

Keyword

nonn,easy

Author

Omar E. Pol, Apr 26 2008

Status

approved