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A057587 Nonnegative numbers of form n*(n^2+-1)/2.

%I A057587 #32 Mar 02 2024 16:36:21
%S A057587 0,1,3,5,12,15,30,34,60,65,105,111,168,175,252,260,360,369,495,505,
%T A057587 660,671,858,870,1092,1105,1365,1379,1680,1695,2040,2056,2448,2465,
%U A057587 2907,2925,3420,3439,3990,4010,4620,4641,5313,5335,6072,6095,6900,6924,7800
%N A057587 Nonnegative numbers of form n*(n^2+-1)/2.
%C A057587 Taking alternate terms gives A027480 and A006003. - _Jeremy Gardiner_, Apr 10 2005
%H A057587 Colin Barker, <a href="/A057587/b057587.txt">Table of n, a(n) for n = 0..1000</a>
%H A057587 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,3,-3,-3,3,1,-1).
%F A057587 a(n) = (2*n^3+9*n^2+15*n+5+(3*n^2+n-5)*(-1)^n)/32. - _Luce ETIENNE_, Nov 18 2014
%F A057587 From _Wesley Ivan Hurt_, Mar 27 2015: (Start)
%F A057587 G.f.: x*(1 + 2 x - x^2 + x^3)/((x - 1)^4*(x + 1)^3).
%F A057587 a(n) = a(n-1)+3*a(n-2)-3*a(n-3)-3*a(n-4)+3*a(n-5)+a(n-6)-a(n-7). (End)
%F A057587 a(2*n+1) + a(2*n) = (a(2*n+1) - a(2*n))^3. - _Greg Dresden_, Feb 10 2022
%F A057587 a(2*n+1)^2 - a(2*n)^2 = (n+1)^4. - _Adam Michael Bere_, Feb 15 2024
%p A057587 A057587:=n->(2*n^3+9*n^2+15*n+5+(3*n^2+n-5)*(-1)^n)/32: seq(A057587(n), n=0..50); # _Wesley Ivan Hurt_, Mar 27 2015
%t A057587 CoefficientList[Series[x*(1 + 2 x - x^2 + x^3)/((x - 1)^4*(x + 1)^3), {x, 0, 50}], x] (* _Wesley Ivan Hurt_, Mar 27 2015 *)
%t A057587 Table[(2 n^3 + 9 n^2 + 15 n + 5 + (3 n^2 + n - 5) (-1)^n) / 32, {n, 0, 50}] (* _Vincenzo Librandi_, Mar 28 2015 *)
%t A057587 LinearRecurrence[{1,3,-3,-3,3,1,-1},{0,1,3,5,12,15,30},50] (* _Harvey P. Dale_, Nov 29 2022 *)
%o A057587 (PARI) concat(0, Vec(x*(x^3-x^2+2*x+1)/((x-1)^4*(x+1)^3) + O(x^100))) \\ _Colin Barker_, Nov 18 2014
%o A057587 (Magma) [(2*n^3+9*n^2+15*n+5+(3*n^2+n-5)*(-1)^n)/32 : n in [0..50]]; // _Wesley Ivan Hurt_, Mar 27 2015
%Y A057587 Cf. A027480, A006003.
%K A057587 nonn,easy
%O A057587 0,3
%A A057587 _N. J. A. Sloane_, Oct 05 2000