A000205 Number of positive integers <= 2^n of form x^2 + 3 y^2. (Formerly M2350 N0928)

1, 1, 3, 4, 8, 14, 25, 45, 82, 151, 282, 531, 1003, 1907, 3645, 6993, 13456, 25978, 50248, 97446, 189291, 368338, 717804, 1400699, 2736534, 5352182, 10478044, 20531668, 40264582, 79022464, 155196838, 304997408, 599752463, 1180027022, 2322950591, 4575114295

Offset

0,3

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

Seth A. Troisi, Table of n, a(n) for n = 0..45

P. Moree and H. J. J. te Riele, The hexagonal versus the square lattice, arXiv:math/0204332 [math.NT], 2002.

P. Moree and H. J. J. te Riele, The hexagonal versus the square lattice, Math. Comp. 73 (2004), no. 245, 451-473.

D. Shanks and L. P. Schmid, Variations on a theorem of Landau. Part I, Math. Comp., 20 (1966), 551-569.

Index entries for sequences related to populations of quadratic forms

Mathematica

(* This program is not suitable to compute more than a score of terms. *)
a[0] = a[1] = 1; a[n_] := a[n] = Module[{cnt, k, r, x, y}, For[cnt = a[n-1]; k = 2^(n-1)+1, k <= 2^n, k++, r = Reduce[x >= 0 && y >= 0 && k == x^2 + 3 y^2, {x, y}, Integers]; If[r =!= False, cnt++]]; cnt];
Table[Print[n, " ", a[n]]; a[n], {n, 0, 20}] (* Jean-François Alcover, Jan 23 2019 *)

Crossrefs

Cf. A000050, A003136.

Sequence in context: A023623 A023558 A170902 * A136425 A331330 A005907

Adjacent sequences: A000202 A000203 A000204 * A000206 A000207 A000208

Keyword

nonn

Author

N. J. A. Sloane

Status

approved