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

Author

Gary W. Adamson, Aug 14 2006

Keywords

• sign
• easy

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

Examples

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

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).

Crossrefs

Cf. A058794, A080663.

Programs

Mathematica

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

PARI

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

Formula

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

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. - Colin Barker, Oct 16 2013

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

Extensions

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