A124011 Add three, add six, add nine, ....

5, 8, 14, 23, 35, 50, 68, 89, 113, 140, 170, 203, 239, 278, 320, 365, 413, 464, 518, 575, 635, 698, 764, 833, 905, 980, 1058, 1139, 1223, 1310, 1400, 1493, 1589, 1688, 1790, 1895, 2003, 2114, 2228, 2345, 2465, 2588, 2714, 2843, 2975, 3110, 3248, 3389, 3533, 3680

Offset: 1

Author

Ruben Fritzon (ruben.fritzon(AT)edu.ovanaker.se), Dec 11 2006

Keywords

• nonn
• easy

Comments

Found on a quiz.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A000217, A016945, A045943.

Programs

Magma

I:=[5,8,14]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..50]]; // Vincenzo Librandi, Feb 15 2012

Mathematica

LinearRecurrence[{3,-3,1},{5,8,14},50] (* Vincenzo Librandi, Feb 15 2012 *)
Table[(3x^2-3x+10)/2,{x,50}] (* Harvey P. Dale, Jul 25 2019 *)
Accumulate[3*Range[0,50]]+5 (* Harvey P. Dale, Jan 16 2024 *)

PARI

a(n)=3*n*(n-1)/2+5 \\ Charles R Greathouse IV, Jun 17 2017

Formula

a(n) = 3*n*(n-1)/2 + 5 = 3*A000217(n-1) + 5 = A045943(n) + 5.

a(n) = 3*n + a(n-1) - 3 with a(1)=5. - Vincenzo Librandi, Nov 28 2009

G.f.: x*(5 - 7*x + 5*x^2)/(1-x)^3. - Colin Barker, Jan 14 2012

3*(8*a(n) - 37) = A016945(n-1)^2. - Vincenzo Librandi, Feb 15 2012

From Elmo R. Oliveira, Feb 11 2025: (Start)

E.g.f.: exp(x)*(5 + 3*x^2/2) - 5.

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

Extensions

More terms from Graham Roy (groy(AT)ashland.edu), Dec 14 2006

Additional comments from Christopher N. Swanson (cswanson(AT)ashland.edu), R. J. Mathar, Dec 14 2006