A139274 a(n) = n*(8*n-1).

0, 7, 30, 69, 124, 195, 282, 385, 504, 639, 790, 957, 1140, 1339, 1554, 1785, 2032, 2295, 2574, 2869, 3180, 3507, 3850, 4209, 4584, 4975, 5382, 5805, 6244, 6699, 7170, 7657, 8160, 8679, 9214, 9765, 10332, 10915, 11514, 12129, 12760, 13407, 14070, 14749

Offset

0,2

Comments

Sequence found by reading the line from 0, in the direction 0, 7, ..., in the square spiral whose vertices are the triangular numbers A000217.

Polygonal number connection: 2*P_n + 5*S_n where P_n is the n-th pentagonal number and S_n is the n-th square. - William A. Tedeschi, Sep 12 2010

Links

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

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

Amelia Carolina Sparavigna, The groupoid of the Triangular Numbers and the generation of related integer sequences, Politecnico di Torino, Italy (2019).

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

Formula

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) - 9 (with a(0)=0). - Vincenzo Librandi, Aug 03 2010

a(n) = (1/3) * Sum_{i=n..(7*n-1)} i. - Wesley Ivan Hurt, Dec 04 2016

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

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

E.g.f.: (8*x^2 + 7*x)*exp(x). (End)

Sum_{n>=1} 1/a(n) = 4*log(2) + sqrt(2)*log(sqrt(2)+1) - (sqrt(2)+1)*Pi/2. - Amiram Eldar, Mar 18 2022

Example

a(1) = 16*1 + 0 - 9 = 7; a(2) = 16*2 + 7 - 9 = 30; a(3) = 16*3 + 30 - 9 = 69. - Vincenzo Librandi, Aug 03 2010

Maple

A139274:=n->n*(8*n-1): seq(A139274(n), n=0..100); # Wesley Ivan Hurt, Dec 04 2016

Mathematica

CoefficientList[Series[x (9 x + 7)/(1 - x)^3, {x, 0, 43}], x] (* Michael De Vlieger, Jan 11 2020 *)
Table[n(8n-1), {n, 0, 50}] (* or *) LinearRecurrence[{3, -3, 1}, {0, 7, 30}, 50] (* Harvey P. Dale, Apr 01 2024 *)

Prog

(Magma) [n*(8*n-1) : n in [0..50]]; // Wesley Ivan Hurt, Dec 04 2016
(PARI) a(n)=n*(8*n-1) \\ Charles R Greathouse IV, Jun 17 2017

Crossrefs

Cf. A000217, A014634, A014635, A033585, A033586, A033587, A035008, A051870, A069129, A085250, A072279, A139272, A139273, A139275, A139276, A139278, A139279, A139280, A139281, A139282.

Sequence in context: A063128 A063148 A116283 * A086640 A083474 A030440

Adjacent sequences: A139271 A139272 A139273 * A139275 A139276 A139277

Keyword

easy,nonn

Author

Omar E. Pol, Apr 26 2008

Status

approved