A078878 Irreducible polynomial coefficient of singular value associated with sqrt(2n).

2, 6, 12, 20, 36, 60, 88, 132, 196, 280, 396, 552, 748, 1020, 1368, 1800, 2376, 3096, 4004, 5160, 6600, 8372, 10584, 13320, 16652, 20760, 25764, 31824, 39204, 48120, 58800, 71688, 87120, 105520, 127512, 153640, 184628, 221364, 264792, 315920

Offset

1,1

Comments

"We conclude this section by touching on the matter of singular values, k_p, which for us are defined to be the solutions in (0,1) of W_p(k'^2,k^2) = 0. These are often called singular moduli for the function lambda. ... Then, since K'(k)/K(k) is isotone, Theorem 2.3(b) shows that this is the unique solution to K'/K(k_p) = root(p) 0 < k_p < 1.

"In the notation of equation (3.2.1), k_p = lambda^*(p) and k'_p=lambda^*(1/p), so that k_p=k(e^{-pi sqrt(p)}) and l_p := k'_p=k(e^{pi/sqrt(p)}). Sophisticated number-theoretic techniques are available for computing k_p for large p, without knowledge of W_p. ... For small p one can solve directly for k_p. Thus k_1 = 1/root(2), k_2 = root(2)-1, k_3 = root(2)(root(3)-1)/4, k_4 = 3-2root(2), ..." - Borwein and Borwein

(K'(x)/K(x))^2 = 2n for some x, whose irreducible polynomial is 1-a(n)x+...

References

J. M. Borwein and P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 139.

Links

Table of n, a(n) for n=1..40.

Example

K'/K = sqrt(6) for k a root of 0 = 1 - 12*x + 2*x^2 + 12*x^3 + x^4 so a(3) = 12.

Crossrefs

Sequence in context: A003274 A259470 A121315 * A095361 A095362 A288984

Adjacent sequences: A078875 A078876 A078877 * A078879 A078880 A078881

Keyword

nonn

Author

Michael Somos, Dec 13 2002

Status

approved