A260481 Field discriminant of n-th composite, f(f(...f(r)...)), where r = 1 and f(x) is the continued fraction [x,x,x, ...].

1, 5, 725, 494613125, 237200374061503726953125, 57083011552674242150083383668890855252740781729278564453125

Offset

0,2

Comments

f(x) = [x,x,x, ...] = (1/2) (x + sqrt((4 + x^2));

f(f(x)) = (1/4)(x + sqrt(4 + x^2)) + (1/2)sqrt(4 + (1/4)(x + sqrt(4 + x^2))^2))/2.

Divisibility conjecture: a(n)^2 | a(n+1), for n>=1; see Example.

Links

Clark Kimberling, Table of n, a(n) for n = 0..7

Example

f(1) = (1 + sqrt(5))/2 = golden ratio;
f(f(1)) = (1 + sqrt(5) + sqrt(22 + 2 sqrt(5)))/4;
D(f(1)) = 5; D(f(f(1))) = 725;
a(2)/(a(1)^2) = 725/5^2 = 29;
a(3)/(a(2)^2) = 941;
a(4)/(a(3)^2) = 969581.
(Regarding n = 0, the zeroth composite of f is taken to be 1.)

Mathematica

s[1] = x; t[1] = 1; z = 8;
s[n_] := s[n] = s[n - 1]^2 - t[n - 1]^2; t[n_] := t[n] = s[n - 1]*t[n - 1];
coeffs[n_] := Apply[Riffle, Map[DeleteCases[#, 0] &, CoefficientList[{s[n], t[n]}, x]]];
polys = Table[Root[Total[Reverse[coeffs[n]] #^(Range[1 + (2^(n - 1))] - 1)] &, 1(*2^(n-1)*)], {n, z}];
m = Map[NumberFieldDiscriminant, polys]  (* Peter J. C. Moses, Jul 30 2015 *)
Table[m[[n + 1]]/m[[n]]^2, {n, 1, z - 1}] (* divisibility conjecture *)

Crossrefs

Cf. A259440, A260457, A260843, A260844.

Sequence in context: A213932 A199089 A345357 * A157281 A171269 A172890

Adjacent sequences: A260478 A260479 A260480 * A260482 A260483 A260484

Keyword

nonn,easy

Author

Clark Kimberling, Aug 13 2015

Status

approved