A153169 a(n) = 4*n^2 + 12*n + 3.

19, 43, 75, 115, 163, 219, 283, 355, 435, 523, 619, 723, 835, 955, 1083, 1219, 1363, 1515, 1675, 1843, 2019, 2203, 2395, 2595, 2803, 3019, 3243, 3475, 3715, 3963, 4219, 4483, 4755, 5035, 5323, 5619, 5923, 6235, 6555, 6883, 7219, 7563, 7915, 8275, 8643, 9019, 9403

Offset: 1

Author

Vincenzo Librandi, Dec 20 2008

Keywords

• nonn
• easy

Comments

Sequence gives values of x such that x^3 + 6*x^2 = y^2 since a(n)^3 + 6*a(n)^2 = (8*n^3 + 36*n^2 + 42*n + 9)^2.

The complete list of nonnegative values of x in x^3 + 6*x^2 = y^2 is given by A028878. - Bruno Berselli, Jan 25 2012

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. A028878, A153167.

Programs

Magma

I:=[19, 43, 75]; [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 25 2012

Mathematica

LinearRecurrence[{3, -3, 1}, {19, 43, 75}, 50] (* Vincenzo Librandi, Feb 25 2012 *)

PARI

for(n=1, 50, print1(4*n^2 + 12*n + 3", ")); \\ Vincenzo Librandi, Feb 25 2012

Formula

From Colin Barker, Jan 24 2012: (Start)

G.f.: x*(19 - 14*x + 3*x^2)/(1-x)^3.

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(1)=19, a(2)=43, a(3)=75. (End)

Sum_{n>=1} 1/a(n) = -2/15 + tan(sqrt(3/2)*Pi)*Pi/(4*sqrt(6)). - Amiram Eldar, Mar 02 2023

From Elmo R. Oliveira, Jun 03 2025: (Start)

E.g.f.: -3 + (3 + 16*x + 4*x^2)*exp(x).

a(n) = A028878(2*n) for n >= 1. (End)

Extensions

Definition rewritten by Bruno Berselli, Jan 25 2012