

A082682


Algebraic degree of R(e^(n * Pi)), where R(q) is the RogersRamanujan continued fraction.


1



8, 4, 32, 8, 40, 16, 64, 16, 96, 20, 96, 32, 96, 32, 160, 32, 128, 48, 160, 40, 256
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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 RogersRamanujan Continued Fraction, The Mathematica Journal, 9:2, 314333, 2004.
Eric Weisstein's World of Mathematics, RogersRamanujan Continued Fraction


EXAMPLE

R(e^(Pi))=Root[114*#1+22*#1^222*#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[12*#16*#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}] (* JeanFrançois 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



