A000692 An approximation to population of x^2 + y^2 <= 2^n. (Formerly M2311 N0913)

1, 3, 4, 5, 9, 15, 27, 50, 92, 171, 322, 610, 1161, 2220, 4260, 8201, 15828, 30622, 59362, 115287, 224260, 436871, 852161, 1664196, 3253531, 6366973, 12471056, 24447507, 47962236, 94161474, 184983976, 363632192, 715220838, 1407510311

Offset

0,2

References

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Table of n, a(n) for n=0..33.

D. Hare, The constant c [Dave Hare, May 21 1996].

D. Shanks, The second-order term in the asymptotic expansion of B(x), Math. Comp., 18 (1964), 75-86.

Index entries for sequences related to populations of quadratic forms.

Formula

a(n) = (b*2^n / sqrt(n*log(2))) * (1 + c/(n*log(2))) where b=0.764223654... is the Landau-Ramanujan constant (A064533) and c=0.5819486593... is the second-order Landau-Ramanujan constant (A227158) given by c = (1/2) * (1-log(Pi*e^gamma/(2*L))) - (1/4) * D(1) where D(s) = (d/ds)(log(Product_{p prime == 3 (mod 4)} 1/(1-p^(-2*s)))) and L is the Lemniscate constant (A064853) [see (12) in Shanks]. - Sean A. Irvine, Feb 25 2011

Crossrefs

Cf. A064533.

Other population sequences for x^2 + y^2: A000050, A000690, A000691.

Sequence in context: A050161 A195609 A117125 * A195971 A080552 A215176

Adjacent sequences: A000689 A000690 A000691 * A000693 A000694 A000695

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from Sean A. Irvine, Feb 24 2011

Name clarified by Seth A. Troisi, May 23 2022

Status

approved