A087205 - OEIS

%I #25 Sep 08 2022 08:45:11

%S 1,2,0,8,-16,64,-192,640,-2048,6656,-21504,69632,-225280,729088,

%T -2359296,7634944,-24707072,79953920,-258736128,837287936,-2709520384,

%U 8768192512,-28374466560,91821703168,-297141272576,961569357824,-3111703805952

%N a(n) = -2*a(n-1) + 4*a(n-2), a(0)=1, a(1)=2.

%C Inverse binomial transform of A087204.

%H G. C. Greubel, <a href="/A087205/b087205.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (-2,4)

%F a(n) = (-1-sqrt(5))^n * (1/2-3*sqrt(5)/10) + (-1+sqrt(5))^n * (1/2+3*sqrt(5)/10).

%F G.f.: (4*x +1)/(-4*x^2 +2*x +1). - _Joerg Arndt_, Jul 14 2013

%F a(n+2) = A085449(n)*(-1)^(n+1); a(n+3) = A063727(n)*(-1)^n.

%F a(n) = -(-2)^n*F(n-2) for n >= 0, with F = A000045, and F(-1) = 1, F(-2) = -1. - _Wolfdieter Lang_, Oct 08 2018

%t Table[-(-2)^n*Fibonacci[n - 2], {n, 0, 50}] (* _G. C. Greubel_, Oct 08 2018 *)

%t LinearRecurrence[{-2,4},{1,2},30] (* _Harvey P. Dale_, Jan 24 2022 *)

%o (PARI) Vec((4*x+1)/(-4*x^2+2*x+1)+O(x^66)) \\ _Joerg Arndt_, Jul 14 2013

%o (PARI) vector(50, n, n--; (-1)^(n+1)*2^n*fibonacci(n-2)) \\ _G. C. Greubel_, Oct 08 2018

%o (Magma) [(-1)^(n+1)*2^n*Fibonacci(n-2): n in [0..50]]; // _G. C. Greubel_, Oct 08 2018

%Y Cf. A000045, A063727, A084222, A085449, A087204.

%K easy,sign

%O 0,2

%A _Paul Barry_, Aug 25 2003