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

Author

Pontus von Brömssen, Dec 10 2024

Keywords

• nonn

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.

Examples

			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.

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.