A134538 a(n) = 5*n^2 - 1.

4, 19, 44, 79, 124, 179, 244, 319, 404, 499, 604, 719, 844, 979, 1124, 1279, 1444, 1619, 1804, 1999, 2204, 2419, 2644, 2879, 3124, 3379, 3644, 3919, 4204, 4499, 4804, 5119, 5444, 5779, 6124, 6479, 6844, 7219, 7604, 7999, 8404, 8819, 9244, 9679, 10124, 10579, 11044

Offset: 1

Author

Artur Jasinski, Oct 30 2007

Keywords

• nonn
• easy

Comments

For k != 0, the quintic polynomials of the form x^5 + 5*(5*k^2 - 1)*x + 4*(5*k^2 - 1) have Galois group A5 (order 60) over rational numbers.

Links

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

Programs

Magma

[5*n^2-1: n in [1..50]]; // Vincenzo Librandi, Jul 09 2012

Mathematica

Table[5n^2 - 1, {n, 1, 50}]
CoefficientList[Series[(4+7*x-x^2)/(1-x)^3,{x,0,50}],x] (* Vincenzo Librandi, Jul 09 2012 *)

PARI

a(n)=5*n^2-1 \\ Charles R Greathouse IV, Jul 01 2013

Formula

G.f.: x*(-4-7*x+x^2)/(-1+x)^3. - R. J. Mathar, Nov 14 2007

From Amiram Eldar, Feb 04 2021: (Start)

Sum_{n>=1} 1/a(n) = (1 - (Pi/sqrt(5))*cot(Pi/sqrt(5)))/2.

Sum_{n>=1} (-1)^(n+1)/a(n) = ((Pi/sqrt(5))*csc(Pi/sqrt(5)) - 1)/2.

Product_{n>=1} (1 + 1/a(n)) = (Pi/sqrt(5))*csc(Pi/sqrt(5)).

Product_{n>=1} (1 - 1/a(n)) = csc(Pi/sqrt(5))*sin(sqrt(2/5)*Pi)/sqrt(2). (End)

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

E.g.f.: 1 + (-1 + 5*x + 5*x^2)*exp(x).

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