A123972 a(n) = n^3 - n^2 - 2*n + 1.

1, -1, 1, 13, 41, 91, 169, 281, 433, 631, 881, 1189, 1561, 2003, 2521, 3121, 3809, 4591, 5473, 6461, 7561, 8779, 10121, 11593, 13201, 14951, 16849, 18901, 21113, 23491, 26041, 28769, 31681, 34783, 38081, 41581, 45289, 49211, 53353, 57721, 62321, 67159, 72241

Offset

0,4

Comments

a(n) is the determinant of the 3 X 3 matrix {{n,-1,0 },{-1,n,-1},{0,-1,n-1}}.

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) = (n + 2*cos(2*Pi/7)) * (n + 2*cos(4*Pi/7)) * (n + 2*cos(6*Pi/7)). Cf. 3rd column from the left in the array of A162997. - Gary W. Adamson, Jul 23 2009

a(n), n>2 equals lower right term in M^3, M is the 2 X 2 matrix {{1,(n-2)}{1,(n-1}}. - Gary W. Adamson, Jun 29 2011

Starting (1, 13, 41, ...) = the binomial transform of (1, 12, 16, 6). - Gary W. Adamson, Jun 29 2011

G.f.: (1 - 5*x + 11*x^2 - x^3)/(1-x)^4. - Colin Barker, Jan 29 2012

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Vincenzo Librandi, Jun 27 2012

Maple

with(linalg): M:=n->matrix(3, 3, [n, -1, 0, -1, n, -1, 0, -1, n-1]): seq(det(M(n)), n=0..42);

Mathematica

CoefficientList[Series[(1-5*x+11*x^2-x^3)/(1-x)^4, {x, 0, 40}], x] (* Vincenzo Librandi, Jun 27 2012 *)

Prog

(PARI) a(n)=n^3-n^2-2*n+1 \\ Charles R Greathouse IV, Jun 30 2011
(Magma) I:=[1, -1, 1, 13]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..50]]; // Vincenzo Librandi, Jun 27 2012

Crossrefs

Cf. A162997.

Sequence in context: A102083 A139866 A026918 * A167585 A141970 A305155

Adjacent sequences: A123969 A123970 A123971 * A123973 A123974 A123975

Keyword

sign,easy

Author

Gary W. Adamson and Roger L. Bagula, Oct 30 2006

Extensions

Edited by N. J. A. Sloane, Nov 01 2006 and Nov 24 2006

Status

approved