A000692 An approximation to population of x^2 + y^2 <= 2^n.

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

Author

N. J. A. Sloane

Keywords

• nonn

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

• 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.

Crossrefs

Cf. A064533.

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

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

Extensions

More terms from Sean A. Irvine, Feb 24 2011

Name clarified by Seth A. Troisi, May 23 2022