A057722 a(n) = n^4 - 3*n^2 + 1.

1, -1, 5, 55, 209, 551, 1189, 2255, 3905, 6319, 9701, 14279, 20305, 28055, 37829, 49951, 64769, 82655, 104005, 129239, 158801, 193159, 232805, 278255, 330049, 388751, 454949, 529255, 612305, 704759, 807301, 920639, 1045505, 1182655

Offset: 0

Author

N. J. A. Sloane, Oct 27 2000

Keywords

• sign
• easy

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Cf. A028387.

Programs

GAP

List([0..40], n-> n^4 -3*n^2 +1); # G. C. Greubel, Aug 12 2019

Magma

[n^4 -3*n^2 +1: n in [0..40]]; // G. C. Greubel, Aug 12 2019

Maple

seq(n^4 -3*n^2 +1, n=0..40); # G. C. Greubel, Aug 12 2019

Mathematica

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 *)
((2*Range[0, 40]^2 -3)^2 -5)/4 (* G. C. Greubel, Aug 12 2019 *)

PARI

vector(40, n, n--; n^4 -3*n^2 +1) \\ G. C. Greubel, Aug 12 2019

Sage

[n^4 -3*n^2 +1 for n in (0..40)] # G. C. Greubel, Aug 12 2019

Formula

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

From G. C. Greubel, Aug 12 2019: (Start)

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

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

a(n) = A028387(n-2)*A028387(n-1). - Lamine Ngom, Oct 27 2020