A051494 - OEIS

%I #25 Sep 08 2022 08:44:59

%S 1,-1,4,-2,9,-3,16,-4,25,-5,36,-6,49,-7,64,-8,81,-9,100,-10,121,-11,

%T 144,-12,169,-13,196,-14,225,-15,256,-16,289,-17,324,-18,361,-19,400,

%U -20,441,-21,484,-22,529,-23,576,-24,625,-25,676,-26,729,-27,784,-28,841,-29,900,-30

%N Expansion of (1 - x + x^2 + x^3)/(1 - x^2)^3.

%C Or, n^2 followed by -n. - _Mohammad K. Azarian_, Aug 29 2005

%C Also, +-A000217(m) + A000217(m-1) = 1+0, -1+0, 3+1, -3+1, 6+3, -6+3, ..., for m > 0. - _Bruno Berselli_, Jun 07 2013

%H Vincenzo Librandi, <a href="/A051494/b051494.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (0,3,0,-3,0,1).

%F a(n) = (n^2 + 2*n + 2 + (n^2 + 6*n + 6)*(-1)^n)/8. - _Bruno Berselli_, Jun 07 2013

%t LinearRecurrence[{0, 3, 0, -3, 0, 1}, {1, -1, 4, -2, 9, -3}, 60] (* _Bruno Berselli_, Jun 07 2013 *)

%t CoefficientList[Series[(1 - x + x^2 + x^3) / (1 - x^2)^3, {x, 0, 40}], x] (* _Vincenzo Librandi_, Jun 19 2013 *)

%o (Magma) /* By the second comment: */ A000217:=func<i | i*(i+1)/2>; [s*A000217(n)+A000217(n-1): s in [1,-1], n in [1..30]]; // _Bruno Berselli_, Jun 07 2013

%K sign,easy

%O 0,3

%A _N. J. A. Sloane_