A088139 a(n) = 2*a(n-1) - 6*a(n-2), a(0)=0, a(1)=1.

0, 1, 2, -2, -16, -20, 56, 232, 128, -1136, -3040, 736, 19712, 35008, -48256, -306560, -323584, 1192192, 4325888, 1498624, -22958080, -54907904, 27932672, 385312768, 603029504, -1105817600, -5829812224, -5024718848, 24929435648, 80007184384

Offset: 0

Author

Paul Barry, Sep 20 2003

Keywords

• easy
• sign

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (2,-6).

Crossrefs

Cf. A045873, A084102, A088136, A088137.

Programs

GAP

a:=[0,1];; for n in [3..30] do a[n]:=2*a[n-1]-6*a[n-2]; od; a; # Muniru A Asiru, Oct 23 2018

Magma

I:=[0,1]; [n le 2 select I[n] else 2*Self(n-1) - 6*Self(n-2): n in [1..30]]; // G. C. Greubel, Oct 22 2018

Maple

seq(coeff(series(x/(1-2*x+6*x^2),x,n+1), x, n), n = 0 .. 30); # Muniru A Asiru, Oct 23 2018

Mathematica

Join[{a=0,b=1},Table[c=2*b-6*a;a=b;b=c,{n,100}]] (* Vladimir Joseph Stephan Orlovsky, Jan 17 2011*)
TrigExpand@Table[(6^(n/2) Sin[n ArcTan[Sqrt[5]]])/Sqrt[5], {n, 0, 20}] (* or *)
Table[Sum[(-5)^k Binomial[n, 2 k + 1], {k, 0, n/2}], {n, 0, 20}] (* Vladimir Reshetnikov, Sep 20 2016 *)
LinearRecurrence[{2,-6},{0,1},40] (* Harvey P. Dale, Nov 22 2024 *)

PARI

x='x+O('x^30); concat([0], Vec(x/(1-2*x+6*x^2))) \\ G. C. Greubel, Oct 22 2018

Sage

[lucas_number1(n,2,6) for n in range(0, 30)] # Zerinvary Lajos, Apr 23 2009

Formula

G.f.: x/(1-2*x+6*x^2).

E.g.f.: exp(x)*sin(sqrt(5)*x)/sqrt(5).

a(n) = ((1+i*sqrt(5))^n-(1-i*sqrt(5))^n)/(2*i*sqrt(5)).

a(n) = Im{(1+i*sqrt(5))^n/sqrt(5)}.

a(n) = Sum_{k=0..floor(n/2)} C(n, 2k+1)(-5)^k.

a(n+1) = (-1)^n*Sum_{k, 0<=k<=n} A172250(n,k)*(-2)^k. - Philippe Deléham, Feb 15 2012