A378872 Discriminant of the minimal polynomial of a number whose continued fraction expansion has periodic part given by the n-th composition (in standard order).

5, 8, 5, 13, 12, 12, 5, 20, 21, 8, 40, 21, 40, 40, 5, 29, 32, 60, 17, 60, 85, 85, 96, 32, 17, 85, 96, 17, 96, 96, 5, 40, 45, 24, 104, 13, 148, 148, 165, 24, 148, 8, 221, 148, 12, 221, 260, 45, 104, 148, 165, 148, 221, 12, 260, 104, 165, 221, 260, 165, 260, 260

Offset

1,1

Comments

Here, the minimal polynomial is required to have integer coefficients with no common divisors.

If two numbers have eventually periodic continued fraction expansions with the same periodic part, the discriminants of their respective minimal polynomials are the same.

Links

Table of n, a(n) for n=1..62.

Formula

a(n) = A378873(n)*A378874(n)^2.

a(A059893(n)) = a(n), since reversing the periodic part of a continued fraction leaves the discriminant unchanged.

a(A139706(n)) = a(n), since a circular shift of the periodic part of a continued fraction leaves the discriminant unchanged.

Example

For n = 6, the 5th composition is (1,2). The value of the continued fraction 1+1/(2+1/(1+1/(2+...))) is (1+sqrt(3))/2, whose minimal polynomial is 2*x^2-2*x-1 with discriminant a(6) = 12.

Crossrefs

Cf. A059893, A066099 (compositions in standard order), A139706, A246903, A246921, A305311, A378873, A378874.

Sequence in context: A073212 A059742 A296486 * A011424 A011495 A136258

Adjacent sequences: A378869 A378870 A378871 * A378873 A378874 A378875

Keyword

nonn,new

Author

Pontus von Brömssen, Dec 10 2024

Status

approved