A000075 Number of positive integers <= 2^n of form 2*x^2 + 3*y^2. (Formerly M1078 N0408)

0, 1, 2, 4, 7, 14, 23, 42, 76, 139, 258, 482, 907, 1717, 3269, 6257, 12020, 23171, 44762, 86683, 168233, 327053, 636837, 1241723, 2424228, 4738426, 9271299, 18157441, 35591647, 69820626, 137068908, 269270450, 529312241, 1041093048, 2048825748, 4034059456

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

Bert Dobbelaere, Table of n, a(n) for n = 0..48

Daniel Shanks and Larry 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

Example

a(3)=4 since 2^3=8 and 2=2*1^2, 3=3*1^2, 5=2*1^2+3*1^2, 8=2*2^2.

Prog

(PARI) a(n)=if(n<0, 0, sum(k=1, 2^n, 0<sum(y=0, sqrtint(k\3), issquare((k-3*y^2)/2))))
(Python)
import math
def A000075(n):
    return len(set([2*x**2+3*y**2 for x in range(1+math.isqrt(2**n//2)) for y in range(1+math.isqrt((2**n-2*x**2)//3)) if 0 < 2*x**2+3*y**2 <= 2**n]))
# Chai Wah Wu, Aug 20 2014

Crossrefs

Cf. A002480.

Sequence in context: A079488 A054160 A034426 * A285903 A048248 A370636

Adjacent sequences: A000072 A000073 A000074 * A000076 A000077 A000078

Keyword

nonn

Author

N. J. A. Sloane

Status

approved