OFFSET
1,1
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).
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
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 *)
PROG
(PARI) g2(n) = for(x=1, n, y=x*(x+1)^2; print1(y", "))
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Cino Hilliard, Feb 09 2006
EXTENSIONS
Name changed by Jon E. Schoenfield, Feb 03 2019
STATUS
approved