A000072 Number of positive integers <= 2^n of form x^2 + 4 y^2. (Formerly M1063 N0400)

1, 1, 2, 4, 7, 12, 22, 41, 72, 137, 254, 476, 901, 1716, 3274, 6286, 12090, 23331, 45140, 87511, 169972, 330752, 644499, 1257523, 2456736, 4804666, 9405749, 18429828, 36141339, 70928099, 139295793, 273741700, 538277486, 1059051586, 2084763319, 4105924366

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..60

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

Formula

a(n) = A000050(n) - A000050(n-1) + A000050(n-2). - Seth A. Troisi, Apr 06 2026

Example

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

Prog

(PARI) a(n)=if(n<0, 0, sum(k=1, 2^n, 0<sum(y=0, sqrtint(k\4), issquare(k-4*y^2))))

Crossrefs

Cf. A000050.

Sequence in context: A002573 A288317 A064492 * A268306 A018179 A190165

Adjacent sequences: A000069 A000070 A000071 * A000073 A000074 A000075

Keyword

nonn

Author

N. J. A. Sloane

Status

approved