A082682 Algebraic degree of R(e^(-n * Pi)), where R(q) is the Rogers-Ramanujan continued fraction.

8, 4, 32, 8, 40, 16, 64, 16, 96, 20, 96, 32, 96, 32, 160, 32, 128, 48, 160, 40, 256

Offset

1,1

Comments

All numbers in this sequence are divisible by 4.

All polynomials are symmetric and reducible in rationals extended by 5^(1/2) and 5^(1/4).

References

Computed by Michael Trott.

Links

Table of n, a(n) for n=1..21.

M. Trott, Modular Equations of the Rogers-Ramanujan Continued Fraction, The Mathematica Journal, 9:2, 314-333, 2004.

Eric Weisstein's World of Mathematics, Rogers-Ramanujan Continued Fraction

Example

R(e^(-Pi))=Root[1-14*#1+22*#1^2-22*#1^3+30*#1^4+22*#1^5+22*#1^6+14*#1^7+#1^8&,4], so a(1)=8.
R(e^(-2*Pi))=Root[1-2*#1-6*#1^2+2*#1^3+#1^4&,3], so a(2)=4.

Mathematica

(* Program not suitable to compute more than a few terms *)
terms = 12; prec = 3000; QP = QPochhammer;
R[q_] := q^(1/5)*QP[q, q^5]*QP[q^4, q^5]/(QP[q^2, q^5]*QP[q^3, q^5]);
a[n_] := N[R[E^(-n Pi)], prec] // RootApproximant // MinimalPolynomial[#, x]& // Exponent[#, x]&;
Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 1, terms}] (* Jean-François Alcover, Dec 22 2017 *)

Crossrefs

Cf. A275713 (degree of R(e^(-prime(n) * Pi))).

Sequence in context: A238163 A213773 A213178 * A279635 A213505 A270232

Adjacent sequences: A082679 A082680 A082681 * A082683 A082684 A082685

Keyword

nonn,more,nice

Author

Eric W. Weisstein, Apr 10 2003

Extensions

a(11)-a(21) computed by Artur Jasinski, Aug 24 2016

Status

approved