A167499 a(n) = n*(n+3)/2 + 6.

6, 8, 11, 15, 20, 26, 33, 41, 50, 60, 71, 83, 96, 110, 125, 141, 158, 176, 195, 215, 236, 258, 281, 305, 330, 356, 383, 411, 440, 470, 501, 533, 566, 600, 635, 671, 708, 746, 785, 825, 866, 908, 951, 995, 1040, 1086, 1133, 1181, 1230, 1280, 1331, 1383, 1436, 1490

Offset: 0

Author

Vincenzo Librandi, Nov 07 2009

Keywords

• nonn
• easy

Comments

Numbers m > 5 such that 8*m - 39 is a square. - Bruce J. Nicholson, Jul 25 2017

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Ângela Mestre and José Agapito, Square Matrices Generated by Sequences of Riordan Arrays, Journal of Integer Sequences, Vol. 22 (2019), Article 19.8.4.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A187710.

Programs

Magma

[n*(n+3)/2+6: n in [0..60]]; // Vincenzo Librandi, Sep 16 2013

Maple

A167499:=n->n*(n+3)/2+6: seq(A167499(n), n=0..100); # Wesley Ivan Hurt, Jul 25 2017

Mathematica

Table[n*(n + 3)/2 + 6, {n, 0, 100}] (* Vladimir Joseph Stephan Orlovsky, Jun 03 2011 *)
CoefficientList[Series[(6 - 10 x + 5 x^2) / (1 - x)^3, {x, 0, 60}], x] (* Vincenzo Librandi, Sep 16 2013 *)
LinearRecurrence[{3,-3,1},{6,8,11},60] (* Harvey P. Dale, Jun 21 2022 *)

PARI

a(n)=n*(n+3)/2+6 \\ Charles R Greathouse IV, Jun 17 2017

Formula

a(n) = n + a(n-1) + 1, with n > 1, a(1) = 8.

From Vincenzo Librandi, Sep 16 2013: (Start)

G.f.: (6 - 10*x + 5*x^2)/(1-x)^3.

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

Sum_{n>=0} 1/a(n) = 2*Pi*tanh(Pi*sqrt(39)/2)/sqrt(39) - 1/5. - Amiram Eldar, Jan 17 2021

From Elmo R. Oliveira, Oct 31 2024: (Start)

E.g.f.: exp(x)*(6 + 2*x + x^2/2).

a(n) = A187710(n+1)/2. (End)