A152015 a(n) = n^3 - n^2 - n.

0, -1, 2, 15, 44, 95, 174, 287, 440, 639, 890, 1199, 1572, 2015, 2534, 3135, 3824, 4607, 5490, 6479, 7580, 8799, 10142, 11615, 13224, 14975, 16874, 18927, 21140, 23519, 26070, 28799, 31712, 34815, 38114, 41615, 45324, 49247, 53390, 57759, 62360

Offset: 0

Author

Vladimir Joseph Stephan Orlovsky, Nov 20 2008

Keywords

• sign
• easy

Comments

For n > 1, these are also the largest positive integers k such that k + n divides k^3 + n^2. For all n > 1 and p > 1, the largest positive integer k such that k + n divides k^p + n^(p-1) is given by k = n^p - (-n)^(p-1) - n. Here, p = 3. - Derek Orr, Aug 13 2014

Links

• Derek Orr, Table of n, a(n) for n = 0..10000
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A002414, A005449.

Programs

Magma

[n^3-n^2-n : n in [0..50]]; // Wesley Ivan Hurt, Aug 13 2014

Maple

a:=n->sum(-1+sum(1+sum(1,i=2..n),j=2..n),k=1..n): seq(a(n), n=0..44); # Zerinvary Lajos, Dec 22 2008
A152015:=n->n^3-n^2-n: seq(A152015(k), k=0..100); # Wesley Ivan Hurt, Oct 06 2013

Mathematica

lst={};Do[AppendTo[lst,n^3-n^2-n],{n,0,5!}];lst
Table[n^3-n^2-n, {n,0,100}] (* Wesley Ivan Hurt, Oct 06 2013 *)
LinearRecurrence[{4,-6,4,-1},{0,-1,2,15},50] (* Harvey P. Dale, Sep 08 2024 *)

PARI

vector(100, n, (n-1)^3-(n-1)^2-(n-1)) \\ Derek Orr, Aug 13 2014

Formula

G.f.: -x*(1-6*x-x^2)/(1-x)^4. - Bruno Berselli, Jul 27 2012

a(n) = A002414(n) - A005449(n). - Wesley Ivan Hurt, Oct 06 2013

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Wesley Ivan Hurt, Aug 13 2014

E.g.f.: exp(x)*x*(x^2 + 2*x - 1). - Stefano Spezia, Apr 15 2022

Extensions

Offset changed by Bruno Berselli, Jul 27 2012