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%I #23 Dec 28 2022 09:04:43
%S 2,-1,1,-1,0,3,-3,0,5,-5,0,7,-7,0,9,-9,0,11,-11,0,13,-13,0,15,-15,0,
%T 17,-17,0,19,-19,0,21,-21,0,23,-23,0,25,-25,0,27,-27,0,29,-29,0,31,
%U -31,0,33,-33,0,35,-35,0,37,-37,0,39,-39
%N First differences of coefficients in the continued fraction for e.
%C First differences of A003417.
%H G. C. Greubel, <a href="/A120691/b120691.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 (-1,-1,1,1,1).
%F G.f.: (1-x)*(2+x+2*x^2-3*x^3-x^4+x^6)/(1-2*x^3+x^6).
%F a(n) = 2*C(0,n) -C(1,n) +2*sin(2*Pi*(n-1)/3)*floor((2*n-1)/3)/sqrt(3). [Sign corrected by _M. F. Hasler_, May 01 2013]
%F a(0)=2, a(1)=-1, for n>0: a(3*n-1) = 2*n-1, a(3*n) = 1-2*n, a(3*n+1) = 0. - _M. F. Hasler_, May 01 2013
%F a(n) = - a(n-1) - a(n-2) + a(n-3) + a(n-4) + a(n-5) for n > 6. - _Chai Wah Wu_, Jul 27 2022
%F a(n) = 0 if n mod 3 = 1, a(n) = (2*n-1)/3 if n mod 3 = 2, a(n) = (3-2*n)/3 otherwise, with a(0) = 2, and a(1) = -1. - _G. C. Greubel_, Dec 28 2022
%t Join[{2},Differences[ContinuedFraction[E,120]]] (* or *) LinearRecurrence[{-1,-1,1,1,1},{2,-1,1,-1,0,3,-3},120] (* _Harvey P. Dale_, Jun 08 2016 *)
%o (PARI) A120691(n)={n<2 && return(2-3*n); n=divrem(n-1,3); if(n[2],-(1+n[1]*2)*(-1)^n[2])} \\ - _M. F. Hasler_, May 01 2013
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 70); Coefficients(R!( (1-x)*(2+x+2*x^2-3*x^3-x^4+x^6)/(1-x^3)^2 )); // _G. C. Greubel_, Dec 28 2022
%o (SageMath)
%o def b(n):
%o if (n%3==1): return 0
%o elif (n%3==2): return (2*n-1)/3
%o else: return (3-2*n)/3
%o def A120691(n): return b(n) + (-1)^n*int(n<2)
%o [A120691(n) for n in range(71)] # _G. C. Greubel_, Dec 28 2022
%Y Cf. A003417, A102899.
%K easy,sign
%O 0,1
%A _Paul Barry_, Jun 27 2006
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