A134547 a(n) = 5*n^2 + 20*n + 4.

29, 64, 109, 164, 229, 304, 389, 484, 589, 704, 829, 964, 1109, 1264, 1429, 1604, 1789, 1984, 2189, 2404, 2629, 2864, 3109, 3364, 3629, 3904, 4189, 4484, 4789, 5104, 5429, 5764, 6109, 6464, 6829, 7204, 7589, 7984, 8389, 8804, 9229, 9664, 10109, 10564, 11029

Offset: 1

Author

Artur Jasinski, Oct 31 2007, Nov 21 2007

Keywords

• nonn
• easy

Comments

Most quintic polynomials x^5 + 5*x*(5*n^2 + 20*n + 4) + 8*(5*n^2 + 20*n + 4) (with the exception of n=0 or -4 when the polynomial is solvable, or n=-2 when it is reducible) have nonsolvable alternating Galois group A_5 (of order 60) over rational numbers.

Links

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

Crossrefs

Cf. A134538.

Programs

Maple

seq(5*n^2+20*n+4,n=1..40); # Robert Israel, May 19 2025

Mathematica

Table[5n^2 + 20n + 4, {n, 1, 30}]

PARI

a(n)=5*n^2+20*n+4 \\ Charles R Greathouse IV, Jun 17 2017

Formula

G.f.: -x*(29-23*x+4*x^2)/(-1+x)^3. - R. J. Mathar, Nov 14 2007

From Elmo R. Oliveira, Jun 04 2025: (Start)

E.g.f.: -4 + (4 + 25*x + 5*x^2)*exp(x).

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n >= 4. (End)