A114364 a(n) = n*(n+1)^2.

4, 18, 48, 100, 180, 294, 448, 648, 900, 1210, 1584, 2028, 2548, 3150, 3840, 4624, 5508, 6498, 7600, 8820, 10164, 11638, 13248, 15000, 16900, 18954, 21168, 23548, 26100, 28830, 31744, 34848, 38148, 41650, 45360, 49284, 53428, 57798, 62400

Offset: 1

Author

Cino Hilliard, Feb 09 2006

Keywords

• easy
• nonn

Comments

Former name was "Numbers k such that k*x^3 + x + 1 is not prime."

Theorem: y = k*x^3 + x + 1 is not prime for k = 4, 18, 48, ..., n*(n+1)^2. Proof: n*(n+1)^2*x^3 + x + 1 = ((n+1)*x + 1)*((n^2+n)*x^2 - n*x + 1). Thus (n+1)*x + 1 divides y. This could possibly be used as a pre-test for compositeness. This sequence is the same as beginning with the third term of A045991.

Links

• Jinyuan Wang, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A045991.

Equals twice A006002.

Programs

Maple

seq(2*binomial(n,2)*n, n=2..40); # Zerinvary Lajos, Apr 25 2007

Mathematica

CoefficientList[Series[(2 (2 + x))/(-1 + x)^4, {x, 0, 38}], x] (* or *)
Array[# (# + 1)^2 &, 39] (* Michael De Vlieger, Feb 03 2019 *)

PARI

g2(n) = for(x=1,n,y=x*(x+1)^2;print1(y","))

Formula

a(n) = n*(n+1)^2.

G.f.: 2 * (2 + x)/(-1 + x)^4. - Michael De Vlieger, Feb 03 2019

From Amiram Eldar, Jan 02 2021: (Start)

Sum_{n>=1} 1/a(n) = 2 - Pi^2/6.

Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/12 +2*log(2) - 2. (End)

E.g.f.: exp(x)*x*(4 + 5*x + x^2). - Stefano Spezia, May 20 2021

Extensions

Name changed by Jon E. Schoenfield, Feb 03 2019