A267081 Coefficient of x^3 in the minimal polynomial of the continued fraction [1^n,2^(1/3),1,1,...], where 1^n means n ones.
-9, 35, -969, -14359, -279261, -4862231, -88270665, -1576950691, -28345226121, -508305487319, -9123426587229, -163697793422935, -2937543639603849, -52711355807057699, -945871877489577801, -16972948054702729111, -304567428780675699165, -5465239154667149397911
Offset: 0
Examples
Let u = 2^(1/3), and let p(n,x) be the minimal polynomial of the number given by the n-th continued fraction: [u,1,1,1,...] has p(0,x) = -5 - 15 x - 6 x^2 - 9 x^3 + 3 x^5 + x^6, so that a(0) = -9. [1,u,1,1,1,...] has p(1,x) = -11 + 45 x - 66 x^2 + 35 x^3 + 6 x^4 - 15 x^5 + 5 x^6, so that a(1) = 35; [1,1,u,1,1,1...] has p(2,x) = 131 - 633 x + 1110 x^2 - 969 x^3 + 456 x^4 - 111 x^5 + 11 x^6, so that a(2) = -969.
Links
- Andrew Howroyd, Table of n, a(n) for n = 0..200
- Index entries for linear recurrences with constant coefficients, signature (10, 135, 135, 10, -1).
Programs
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Mathematica
u[n_] := Table[1, {k, 1, n}]; t[n_] := Join[u[n], {2^(1/3)}, {{1}}]; f[n_] := FromContinuedFraction[t[n]]; t = Table[MinimalPolynomial[f[n], x], {n, 0, 30}] Coefficient[t, x, 0]; (* A267078 *) Coefficient[t, x, 1]; (* A267079 *) Coefficient[t, x, 2]; (* A267080 *) Coefficient[t, x, 3]; (* A267081 *) Coefficient[t, x, 4]; (* A267082 *) Coefficient[t, x, 5]; (* A267083 *) Coefficient[t, x, 6]; (* A266527 *) -
PARI
Vec((-9 + 125*x - 104*x^2 - 8179*x^3 - 9491*x^4 - 700*x^5 + 70*x^6)/((1 + x)*(1 + 7*x + x^2)*(1 - 18*x + x^2)) + O(x^30)) \\ Andrew Howroyd, Mar 07 2018
Formula
a(n) = 13*a(n-1) + 104*a(n-2) - 260*a(n-3) - 260*a(n-4) + 104*a(n-5) + 13*a(n-6) - a(n-7) for n > 8.
G.f.: (-9 + 125 x - 104 x^2 - 8179 x^3 - 9491 x^4 - 700 x^5 + 70 x^6)/(1 - 10 x - 135 x^2 - 135 x^3 - 10 x^4 + x^5).
From Andrew Howroyd, Mar 07 2018: (Start)
a(n) = 10*a(n-1) + 135*a(n-2) + 135*a(n-3) + 10*a(n-4) - a(n-5) for n > 6.
G.f.: (-9 + 125*x - 104*x^2 - 8179*x^3 - 9491*x^4 - 700*x^5 + 70*x^6)/((1 + x)*(1 + 7*x + x^2)*(1 - 18*x + x^2)).
(End)
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