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

%I A123972 #43 Sep 17 2024 04:04:49
%S A123972 1,-1,1,13,41,91,169,281,433,631,881,1189,1561,2003,2521,3121,3809,
%T A123972 4591,5473,6461,7561,8779,10121,11593,13201,14951,16849,18901,21113,
%U A123972 23491,26041,28769,31681,34783,38081,41581,45289,49211,53353,57721,62321,67159,72241
%N A123972 a(n) = n^3 - n^2 - 2*n + 1.
%C A123972 a(n) is the determinant of the 3 X 3 matrix {{n,-1,0 },{-1,n,-1},{0,-1,n-1}}.
%H A123972 Vincenzo Librandi, <a href="/A123972/b123972.txt">Table of n, a(n) for n = 0..1000</a>
%H A123972 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A123972 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
%F A123972 a(n) equals the 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
%F A123972 Starting (1, 13, 41, ...) = the binomial transform of (1, 12, 16, 6). - _Gary W. Adamson_, Jun 29 2011
%F A123972 G.f.: (1 - 5*x + 11*x^2 - x^3)/(1-x)^4. - _Colin Barker_, Jan 29 2012
%F A123972 a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - _Vincenzo Librandi_, Jun 27 2012
%p A123972 with(linalg): M:=n->matrix(3,3,[n,-1,0,-1,n,-1,0,-1,n-1]): seq(det(M(n)),n=0..42);
%t A123972 CoefficientList[Series[(1-5*x+11*x^2-x^3)/(1-x)^4,{x,0,40}],x] (* _Vincenzo Librandi_, Jun 27 2012 *)
%o A123972 (PARI) a(n)=n^3-n^2-2*n+1 \\ _Charles R Greathouse IV_, Jun 30 2011
%o A123972 (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
%Y A123972 Cf. A162997.
%K A123972 sign,easy
%O A123972 0,4
%A A123972 _Gary W. Adamson_ and _Roger L. Bagula_, Oct 30 2006
%E A123972 Edited by _N. J. A. Sloane_, Nov 01 2006 and Nov 24 2006