A121670 a(n) = n^3 - 3*n.

0, -2, 2, 18, 52, 110, 198, 322, 488, 702, 970, 1298, 1692, 2158, 2702, 3330, 4048, 4862, 5778, 6802, 7940, 9198, 10582, 12098, 13752, 15550, 17498, 19602, 21868, 24302, 26910, 29698, 32672, 35838, 39202, 42770, 46548, 50542, 54758, 59202, 63880, 68798, 73962

Offset

0,2

Comments

Previous name was: Real part of (n + i)^3, companion to A080663.

Reversing the order of terms in (n + i)^3 to (1 + ni)^3 generates the terms of A080663. E.g, A080663(4) = 47 since (1 + 4i)^3 = (-47 - 52i). Or, (n + i)^3 = (a(n) + A080663(a)i) and (1 + ni)^3 = (-A080663(n) - a(n)i).

Also, numbers n such that the polynomial x^6 - n*x^3 + 1 is reducible. - Ralf Stephan, Oct 24 2013

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Formula

a(n) = Re((n + i)^3).

From Colin Barker, Oct 16 2013: (Start)

a(n) = n^3 - 3*n.

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).

G.f.: -2*x*(x^2-5*x+1)/(x-1)^4. (End)

a(n)^2 = A028872(n)^3 + 3*A028872(n)^2 for n>1. - Bruno Berselli, May 03 2018

a(n) = A058794(n-2) for n>1. - Altug Alkan, May 03 2018

From Elmo R. Oliveira, Jun 05 2026: (Start)

E.g.f.: exp(x)*x*(-2 + 3*x + x^2).

a(n) = A000578(n) - A008585(n).

a(n) = 2*A154560(n-2) for n >= 2. (End)

Example

a(4) = 52 since (4 + i)^3 = (52 + 47i); where 47 = A080663(4).

Mathematica

CoefficientList[Series[-2 x (x^2 - 5 x + 1)/(x - 1)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Jun 11 2014 *)
(* Alternative: *)
Table[n^3-3n, {n, 0, 60}] (* Harvey P. Dale, Nov 30 2021 *)
(* Alternative: *)
LinearRecurrence[{4, -6, 4, -1}, {0, -2, 2, 18}, 60] (* Harvey P. Dale, Nov 30 2021 *)

Prog

(PARI) Vec(-2*x*(x^2-5*x+1)/(x-1)^4 + O(x^100)); \\ Colin Barker, Oct 16 2013

Crossrefs

Cf. A000578, A008585, A028872, A058794, A080663, A154560.

Sequence in context: A213271 A096190 A136434 * A001183 A201124 A173785

Adjacent sequences: A121667 A121668 A121669 * A121671 A121672 A121673

Keyword

sign,easy

Author

Gary W. Adamson, Aug 14 2006

Extensions

Terms corrected, new name, and more terms from Colin Barker, Oct 16 2013

Status

approved