A057722 - OEIS

%I #31 Oct 21 2022 21:09:21

%S 1,-1,5,55,209,551,1189,2255,3905,6319,9701,14279,20305,28055,37829,

%T 49951,64769,82655,104005,129239,158801,193159,232805,278255,330049,

%U 388751,454949,529255,612305,704759,807301,920639,1045505,1182655

%N a(n) = n^4 - 3*n^2 + 1.

%H G. C. Greubel, <a href="/A057722/b057722.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).

%F a(0)=1, a(1)=-1, a(2)=5, a(3)=55, a(4)=209, a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). - _Harvey P. Dale_, Nov 22 2012

%F From _G. C. Greubel_, Aug 12 2019: (Start)

%F G.f.: (1 -6*x +20*x^2 +10*x^3 -x^4)/(1-x)^5.

%F E.g.f.: (1 -2*x +4*x^2 +6*x^3 +x^4)*exp(x). (End)

%F a(n) = A028387(n-2)*A028387(n-1). - _Lamine Ngom_, Oct 27 2020

%p seq(n^4 -3*n^2 +1, n=0..40); # _G. C. Greubel_, Aug 12 2019

%t Table[n^4-3n^2+1,{n,0,40}] (* or *) LinearRecurrence[{5,-10,10,-5,1},{1,-1,5,55,209},40] (* _Harvey P. Dale_, Nov 22 2012 *)

%t ((2*Range[0, 40]^2 -3)^2 -5)/4 (* _G. C. Greubel_, Aug 12 2019 *)

%o (PARI) vector(40, n, n--; n^4 -3*n^2 +1) \\ _G. C. Greubel_, Aug 12 2019

%o (Magma) [n^4 -3*n^2 +1: n in [0..40]]; // _G. C. Greubel_, Aug 12 2019

%o (Sage) [n^4 -3*n^2 +1 for n in (0..40)] # _G. C. Greubel_, Aug 12 2019

%o (GAP) List([0..40], n-> n^4 -3*n^2 +1); # _G. C. Greubel_, Aug 12 2019

%Y Cf. A028387.

%K sign,easy

%O 0,3

%A _N. J. A. Sloane_, Oct 27 2000