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Expansion of (1-x)/(1-2*x^2-x^3).
4

%I #21 Sep 08 2022 08:45:08

%S 1,-1,2,-1,3,0,5,3,10,11,23,32,57,87,146,231,379,608,989,1595,2586,

%T 4179,6767,10944,17713,28655,46370,75023,121395,196416,317813,514227,

%U 832042,1346267,2178311,3524576,5702889,9227463,14930354,24157815,39088171,63245984,102334157

%N Expansion of (1-x)/(1-2*x^2-x^3).

%H G. C. Greubel, <a href="/A078024/b078024.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (0,2,1).

%F a(n) = Fibonacci(n+2) - Lucas(n) + 2*(-1)^n.

%F a(n) = (-1)^n*A112469(n). - _Philippe Deléham_, Apr 19 2013

%F a(n) = A008346(n) - A008346(n-1), n>=1. - _Philippe Deléham_, Apr 19 2013

%F a(n) = Fibonacci(n-2) + 2*(-1)^n. - _Philippe Deléham_, Apr 19 2013

%t LinearRecurrence[{0,2,1},{1,-1,2},50] (* _Harvey P. Dale_, Jan 14 2015 *)

%t Table[Fibonacci[n-2] +2*(-1)^n, {n,0,50}] (* _G. C. Greubel_, Aug 04 2019 *)

%o (PARI) Vec((1-x)/(1-2*x^2-x^3)+O(x^50)) \\ _Charles R Greathouse IV_, Sep 26 2012

%o (Magma) [Fibonacci(n-2) +2*(-1)^n: n in [0..50]]; // _G. C. Greubel_, Aug 04 2019

%o (Sage) [fibonacci(n-2) + 2*(-1)^n for n in (0..50)] # _G. C. Greubel_, Aug 04 2019

%o (GAP) List([0..50], n-> Fibonacci(n-2) + 2*(-1)^n); # _G. C. Greubel_, Aug 04 2019

%Y Cf. A000032, A000045, A008346, A112469.

%K sign,easy

%O 0,3

%A _N. J. A. Sloane_, Nov 17 2002