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A000021 Number of positive integers <= 2^n of form x^2 + 12 y^2.

%I A000021 M0357 N0134 #33 Jul 08 2023 01:18:25
%S A000021 1,1,2,2,6,9,17,30,54,98,183,341,645,1220,2327,4451,8555,16489,31859,
%T A000021 61717,119779,232919,453584,884544,1727213,3376505,6607371,12942012,
%U A000021 25371540,49777187,97731027,192010355,377475336,742512992,1461352025,2877572478,5668965407
%N A000021 Number of positive integers <= 2^n of form x^2 + 12 y^2.
%D A000021 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A000021 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A000021 Delbert L. Johnson, <a href="/A000021/b000021.txt">Table of n, a(n) for n = 0..45</a>
%H A000021 D. Shanks and L. P. Schmid, <a href="http://dx.doi.org/10.1090/S0025-5718-1966-0210678-1">Variations on a theorem of Landau. Part I</a>, Math. Comp., 20 (1966), 551-569.
%H A000021 <a href="/index/Qua#quadpop">Index entries for sequences related to populations of quadratic forms</a>
%e A000021 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.
%o A000021 (PARI) a(n)=if(n<0,0,sum(k=1,2^n,0<sum(y=0,sqrtint(k\12),issquare(k-12*y^2))))
%o A000021 (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))
%o A000021 (Haskell)
%o A000021 a000021 n = length [() | k <- [1..2^n],
%o A000021         sum [a010052 (k - 12*y^2) | y <- [0..a000196 (k `div` 12)]] > 0]
%o A000021 -- _Reinhard Zumkeller_, Apr 16 2012
%Y A000021 Cf. A000196, A010052, A272933.
%K A000021 nonn
%O A000021 0,3
%A A000021 _N. J. A. Sloane_
%E A000021 More terms from _David W. Wilson_, Feb 07 2000