A273319 a(n) = ((2*n+1)^(n+1) + (-1)^n)/(n+1)^2.

2, 2, 14, 150, 2362, 49210, 1280582, 40045166, 1464047858, 61310662578, 2894855376382, 152184891889030, 8817255144288554, 558260148630165098, 38351949989325264182, 2841496569324942436830

Offset: 0

Author

Gionata Neri, May 19 2016

Keywords

• nonn
• easy

Comments

When searching for the smallest k such that n^k + 1 is not squarefree, I noticed that if n is even then n^(n+1) + 1 is not squarefree, and if n is of the form 4*j + 1 (j>0) then n^((n+1)/2) + 1 is not squarefree.

Crossrefs

Cf. A081215.

Programs

Mathematica

Table[((2n + 1)^(n + 1) + (-1)^n)/(n + 1)^2, {n, 0, 15}] (* Alonso del Arte, May 19 2016 *)

Formula

a(n) = Sum_{k = 0..n} (-1)^(n - k)*2^(k+1)*(n+1)^(k-1)*C(n+1, n - k).