A000021 Number of positive integers <= 2^n of form x^2 + 12 y^2. (Formerly M0357 N0134)

1, 1, 2, 2, 6, 9, 17, 30, 54, 98, 183, 341, 645, 1220, 2327, 4451, 8555, 16489, 31859, 61717, 119779, 232919, 453584, 884544, 1727213, 3376505, 6607371, 12942012, 25371540, 49777187, 97731027, 192010355, 377475336, 742512992, 1461352025, 2877572478, 5668965407

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

Delbert L. Johnson, Table of n, a(n) for n = 0..45

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

Example

a(4)=6 since 2^4=16 and 1=1^2, 4=2^2, 9=3^2, 12=12*1^2, 13=1^2+12*1^2, 16=4^2.

Prog

(PARI) a(n)=if(n<0, 0, sum(k=1, 2^n, 0<sum(y=0, sqrtint(k\12), issquare(k-12*y^2))))
(PARI) a(n)=local(A); if(n<0, 0, A=qfrep([1, 0; 0, 12], 2^n); sum(k=1, 2^n, A[k]!=0))
(Haskell)
a000021 n = length [() | k <- [1..2^n],
        sum [a010052 (k - 12*y^2) | y <- [0..a000196 (k `div` 12)]] > 0]
-- Reinhard Zumkeller, Apr 16 2012

Crossrefs

Cf. A000196, A010052, A272933.

Sequence in context: A021819 A359896 A339426 * A367718 A000022 A034805

Adjacent sequences: A000018 A000019 A000020 * A000022 A000023 A000024

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from David W. Wilson, Feb 07 2000

Status

approved