A266808 - OEIS

A266808

Coefficient of x in the minimal polynomial of the continued fraction [1^n,sqrt(2)+sqrt(3),1,1,...], where 1^n means n ones.

-168, -560, -101124, -3288624, -180132168, -7998247028, -384048485640, -17892957477264, -843263161727364, -39567408316416848, -1859687400468342888, -87350263553726629620, -4103880417768964672104, -192790045902230868971504, -9057117701582885083841028

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,1

COMMENTS

See A265762 for a guide to related sequences.

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..595

Index entries for linear recurrences with constant coefficients, signature (34, 714, -4641, -12376, 12376, 4641, -714, -34, 1).

FORMULA

a(n) = 34*a(n-1) + 714*a(n-2) - 4641*a(n-3) - 12376*a(n-4) + 12376*a(n-5) + 4641*a(n-6) - 714*a(n-7) - 34*a(n-8) + a(n-9).

G.f.: -((4 (-42 + 1288 x + 9467 x^2 - 57564 x^3 - 198636 x^4 + 39086 x^5 - 5774 x^6 - 48 x^7 + 3 x^8))/(-1 + 34 x + 714 x^2 - 4641 x^3 - 2376 x^4 + 12376 x^5 + 4641 x^6 - 714 x^7 - 34 x^8 + x^9)).

EXAMPLE

Let u = sqrt(2) and v = sqrt(3), and let p(n,x) be the minimal polynomial of the number given by the n-th continued fraction:

[u+v,1,1,1,...] has p(0,x) = 49 - 168 x - 50 x^2 + 212 x^3 + 47 x^4 - 68 x^5 - 18 x^6 + 4 x^7 + x^8, so that a(0) = -168.

[1,u+v,1,1,1,...] has p(1,x) = 49 - 560 x + 2498 x^2 - 5760 x^3 + 7547 x^4 - 5760 x^5 + 2498 x^6 - 560 x^7 + 49 x^8, so that a(1) = -560;

[1,1,u+v,1,1,1...] has p(2,x) = 25281 - 101124 x + 173262 x^2 - 165852 x^3 + 96847 x^4 - 35252 x^5 + 7790 x^6 - 952 x^7 + 49 x^8, so that a(2) = -101124.

MATHEMATICA

u[n_] := Table[1, {k, 1, n}]; t[n_] := Join[u[n], {Sqrt[2] + Sqrt[3]}, {{1}}];

f[n_] := FromContinuedFraction[t[n]];

t = Table[MinimalPolynomial[f[n], x], {n, 0, 40}];

Coefficient[t, x, 0]; (* A266803 *)

Coefficient[t, x, 1]; (* A266808 *)

Coefficient[t, x, 2]; (* A267061 *)

Coefficient[t, x, 3]; (* A267062 *)

Coefficient[t, x, 4]; (* A267063 *)

Coefficient[t, x, 5]; (* A267064 *)

Coefficient[t, x, 6]; (* A267065 *)

Coefficient[t, x, 7]; (* A267066 *)