A266708 Coefficient of x in minimal polynomial of the continued fraction [1^n,tau,1,1,1,...], where 1^n means n ones and tau = golden ratio = (1 + sqrt(5))/2.
0, -10, -18, -56, -138, -370, -960, -2522, -6594, -17272, -45210, -118370, -309888, -811306, -2124018, -5560760, -14558250, -38114002, -99783744, -261237242, -683927970, -1790546680, -4687712058, -12272589506, -32130056448, -84117579850, -220222683090
Offset: 0
Examples
Let p(n,x) be the minimal polynomial of the number given by the n-th continued fraction: [tau,1,1,1,1,...] = sqrt(5) has p(0,x) = -5 + x^2, so a(0) = 0; [1,tau,1,1,1,...] = (5 + sqrt(5))/5 has p(1,x) = 4 - 10*x + 5*x^2, so a(1) = -10; [1,1,tau,1,1,...] = (9 - sqrt(5))/4 has p(2,x) = 19 - 18*x + 4*x^2, so a(2) = -18.
Links
- Colin Barker, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (2,2,-1).
Programs
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Mathematica
u[n_] := Table[1, {k, 1, n}]; t[n_] := Join[u[n], {GoldenRatio}, {{1}}]; f[n_] := FromContinuedFraction[t[n]]; t = Table[MinimalPolynomial[f[n], x], {n, 0, 20}] Coefficient[t, x, 0] (* A266707 *) Coefficient[t, x, 1] (* A266708 *) Coefficient[t, x, 2] (* A266707 *) -
PARI
a(n) = round((2^(1-n)*(3*(-1)^n*2^(1+n)+(3-sqrt(5))^n*(-3+2*sqrt(5))-(3+sqrt(5))^n*(3+2*sqrt(5))))/5) \\ Colin Barker, Sep 30 2016
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PARI
concat(0, Vec(-2*x*(5-x)/((1+x)*(1-3*x+x^2)) + O(x^30))) \\ Colin Barker, Sep 30 2016
Formula
G.f.: 2*x*(-5 + x)/((1 + x)*(1 - 3*x + x^2)).
a(n) = 2*a(n-1) - 2*a(n-2) + a(n-3).
a(n) = -2*A192914(n+1).
a(n) = (2^(1-n)*(3*(-1)^n*2^(1+n)+(3-sqrt(5))^n*(-3+2*sqrt(5))-(3+sqrt(5))^n*(3+2*sqrt(5))))/5. - Colin Barker, Sep 30 2016
Comments