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A228936 Expansion of (1+3*x-3*x^3-x^4)/(1+2*x^2+x^4). 2

%I

%S 1,3,-2,-9,2,15,-2,-21,2,27,-2,-33,2,39,-2,-45,2,51,-2,-57,2,63,-2,

%T -69,2,75,-2,-81,2,87,-2,-93,2,99,-2,-105,2,111,-2,-117,2,123,-2,-129,

%U 2,135,-2,-141,2,147

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

%C Optimal simple continued fraction (with signed denominators) of exp(1/3). See A228935.

%C The convergents are a subset of those of the standard regular continued fraction; the sequence of the signs of the difference between the convergents and exp(1/3) starts with: -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, 1, -1, -1, 1, ...

%C For every couple of successive equal signs in this sequence there is a convergent of the standard expansion not present in this one.

%C Repeating the expansion for other numbers of type 1/k a common pattern seems to emerge. Examples:

%C exp(1/4) gives 1, 4, -2, -12, 2, 20, -2, -28, 2, 36, -2, -44, 2, 52, ...

%C exp(1/5) gives 1, 5, -2, -15, 2, 25, -2, -35, 2, 45, -2, -55, 2, 65, ...

%C so it seems that in general the terms for exp(1/k) are generated by the formulas a(0)=1 , a(2n+1)=(-1)^n*k*(2n+1) for n>=0 , a(2n)=(-1)^n*2 for n>0 . These formulas give this expansion for exp(1/k):

%C exp(1/k) = 1+1/(k+1/(-2+1/(-3k+1/(2+1/(5k+1/(-2+1/(-7k+1/(2+....)))))))).

%C that can be rewritten in this equivalent form:

%C exp(1/k) = 1+1/(k-1/(2+1/(3k-1/(2+1/(5k-1/(2+1/(7k-1/(2+....)))))))).

%C This general expansion seems to be valid for any real value of k.

%C Closed form for the general case exp(1/k): b(n) = (1+(-1)^n-(1-(-1)^n)*k*n/2)*i^(n*(n+1)) for n>0 and with i=sqrt(-1). [_Bruno Berselli_, Nov 01 2013]

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

%F This sequence can be generated by these formulas:

%F a(0)=1; for n>=0, a(2n+1) = 3*(-1)^n*(2n+1), a(2n) = 2*(-1)^n for n>0.

%F Formulae for the general case exp(1/k):

%F b(0)=1; for n>=0, b(2n+1) = (-1)^n*k*(2n+1), b(2n) = 2*(-1)^n.

%F b(n) = 2*cos(n*Pi/2)+k*n*sin(n*Pi/2) for n>0.

%F exp(1/k) = 1+1/(k-1/(2+1/(3k-1/(2+1/(5k-1/(2+1/(7k-1/(2+....)))))))).

%F G.f. : (1-x)*(1+x)*(1+k*x+x^2)/(1+x^2)^2.

%F From _Colin Barker_, Oct 26 2013: (Start)

%F a(n) = (-i)^n+i^n+1/2*(((-i)^n-i^n)*n)*(3*i) for n>0, where i=sqrt(-1).

%F a(2n) = 2*(3*n*sin(Pi*n)+cos(Pi*n)) for n>0.

%F a(2n+1) = (6*n+3)*cos(Pi*n)-2*sin(Pi*n) for n>=0.

%F a(n) = -2*a(n-2)-a(n-4) for n>4.

%F G.f.: -(x-1)*(x+1)*(x^2+3*x+1) / (x^2+1)^2. (End)

%e exp(1/3)=1+1/(3+1/(-2+1/(-9+1/(2+1/(15+1/(-2+1/(-21+1/(2+....)))))))) or

%e exp(1/3)=1+1/(3-1/(2+1/(9-1/(2+1/(15-1/(2+1/(21-1/(2+....))))))))

%p SCF := proc (n, q::posint)::list; local L, i, z; Digits := 10000; L := [round(n)]; z := n; for i from 2 to q do if z = op(-1, L) then break end if; z := 1/(z-op(-1, L)); L := [op(L), round(z)] end do; return L end proc

%p SCF(exp(1/3), 50) # _Giovanni Artico_ , Oct 26 2013

%o (PARI) Vec(-(x-1)*(x+1)*(x^2+3*x+1)/(x^2+1)^2+O(x^100)) \\ _Colin Barker_, Oct 26 2013

%Y Cf. A133593, A133570, A228935.

%K sign,cofr,easy

%O 0,2

%A _Giovanni Artico_, Oct 26 2013

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Last modified April 25 00:29 EDT 2019. Contains 322446 sequences. (Running on oeis4.)