A139277 a(n) = n*(8*n+5).

0, 13, 42, 87, 148, 225, 318, 427, 552, 693, 850, 1023, 1212, 1417, 1638, 1875, 2128, 2397, 2682, 2983, 3300, 3633, 3982, 4347, 4728, 5125, 5538, 5967, 6412, 6873, 7350, 7843, 8352, 8877, 9418, 9975, 10548, 11137, 11742, 12363, 13000

Offset

0,2

Comments

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.

Links

Table of n, a(n) for n=0..40.

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 + 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) = 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) - 3 for n > 0, a(0)=0. - Vincenzo Librandi, Aug 03 2010

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

E.g.f.: exp(x)*x*(13 + 8*x). - Elmo R. Oliveira, Dec 15 2024

Mathematica

Table[n (8 n + 5), {n, 0, 50}] (* Bruno Berselli, Aug 22 2018 *)
LinearRecurrence[{3, -3, 1}, {0, 13, 42}, 50] (* Harvey P. Dale, Dec 04 2018 *)

Prog

(PARI) a(n)=n*(8*n+5) \\ Charles R Greathouse IV, Jun 17 2017

Crossrefs

Cf. A000217, A014634, A014635, A033585, A033586, A033587, A035008, A051870, A069129, A085250.

Cf. A139271, A139272, A139273, A139274, A139275, A139276, A139278, A139279, A139280, A139281, A139282.

Sequence in context: A002673 A081301 A159527 * A162265 A233298 A243028

Adjacent sequences: A139274 A139275 A139276 * A139278 A139279 A139280

Keyword

nonn,easy

Author

Omar E. Pol, Apr 26 2008

Status

approved