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A000021 Number of positive integers <= 2^n of form x^2 + 12 y^2.
(Formerly M0357 N0134)
2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

REFERENCES

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

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

Table of n, a(n) for n=0..36.

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.

Sequence in context: A231137 A188808 A021819 * A000022 A034805 A192659

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

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Last modified December 10 17:35 EST 2016. Contains 279005 sequences.