A152950 a(n) = 3 + n*(n-1)/2.

3, 4, 6, 9, 13, 18, 24, 31, 39, 48, 58, 69, 81, 94, 108, 123, 139, 156, 174, 193, 213, 234, 256, 279, 303, 328, 354, 381, 409, 438, 468, 499, 531, 564, 598, 633, 669, 706, 744, 783, 823, 864, 906, 949, 993, 1038, 1084, 1131, 1179, 1228, 1278, 1329, 1381, 1434, 1488

Offset: 1

Author

Vladimir Joseph Stephan Orlovsky, Dec 15 2008

Keywords

• nonn
• easy

Comments

a(1)=3; then add 1 to the first number, then 2, 3, 4, ... and so on.

Numbers m such that 8*m - 23 is a square. - Bruce J. Nicholson, Jul 25 2017

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000
• Ângela Mestre and José Agapito, Square Matrices Generated by Sequences of Riordan Arrays, J. Int. Seq., Vol. 22 (2019), Article 19.8.4.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A000124, A000217, A027691, A152947, A152948, A152949.

Programs

Magma

[3+n*(n-1)/2 : n in [1..50]]; // Wesley Ivan Hurt, Mar 25 2020

Maple

A152950:=n->3 + n*(n-1)/2; seq(A152950(n), n=1..100); # Wesley Ivan Hurt, Jan 28 2014

Mathematica

s=3;lst={3};Do[s+=n;AppendTo[lst,s],{n,1,5!}];lst
Table[3 + n*(n-1)/2, {n, 100}] (* Wesley Ivan Hurt, Jan 28 2014 *)

PARI

a(n)=3+n*(n-1)/2 \\ Charles R Greathouse IV, Oct 07 2015

Sage

[3+binomial(n,2) for n in range(1, 55)] # Zerinvary Lajos, Mar 12 2009

Formula

a(n) = A152949(n+1) = 3 + A000217(n-1). - R. J. Mathar, Jan 03 2009

a(n) = 3 + C(n,2), n >= 1. - Zerinvary Lajos, Mar 12 2009

a(n) = a(n-1) + n - 1 (with a(1)=3). - Vincenzo Librandi, Nov 27 2010

Sum_{n>=1} 1/a(n) = 2*Pi*tanh(sqrt(23)*Pi/2)/sqrt(23). - Amiram Eldar, Dec 13 2022

From Elmo R. Oliveira, Nov 18 2024: (Start)

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

E.g.f.: exp(x)*(3 + x^2/2) - 3.

a(n) = A027691(n-1)/2.

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