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A198772 Numbers having exactly one representation by the quadratic form x^2 + xy + y^2 with 0 <= x <= y.

%I A198772 #17 Jun 21 2018 08:35:22
%S A198772 0,1,3,4,7,9,12,13,16,19,21,25,27,28,31,36,37,39,43,48,52,57,61,63,64,
%T A198772 67,73,75,76,79,81,84,93,97,100,103,108,109,111,112,117,121,124,127,
%U A198772 129,139,144,148,151,156,157,163,171,172,175,181,183,189,192,193
%N A198772 Numbers having exactly one representation by the quadratic form x^2 + xy + y^2 with 0 <= x <= y.
%H A198772 Reinhard Zumkeller, <a href="/A198772/b198772.txt">Table of n, a(n) for n = 1..10000</a>
%F A198772 A088534(a(n)) = 1.
%F A198772 a(n) = A034022(n) for n <= 32.
%e A198772 a(20) = 48 = 4^2 + 4*4 + 4^2, A088534(48) = 1;
%e A198772 a(21) = 52 = 2^2 + 2*6 + 6^2, A088534(52) = 1.
%t A198772 amax = 200; xmax = Sqrt[amax] // Ceiling; Clear[f]; f[_] = 0; Do[q = x^2 + x y + y^2; f[q] = f[q] + 1, {x, 0, xmax}, {y, x, xmax}];
%t A198772 A198772 = Select[Range[0, 3 xmax^2], # <= amax && f[#] == 1&] (* _Jean-François Alcover_, Jun 21 2018 *)
%o A198772 (Haskell)
%o A198772 a198772 n = a198772_list !! (n-1)
%o A198772 a198772_list = filter ((== 1) . a088534) a003136_list
%o A198772 (Julia)
%o A198772 function isA198772(n)
%o A198772     M = Int(round(2*sqrt(n/3)))
%o A198772     count = 0
%o A198772     for y in 0:M, x in 0:y
%o A198772         n == x^2 + y^2 + x*y && (count += 1)
%o A198772         count == 2 && break
%o A198772     end
%o A198772     return count == 1
%o A198772 end
%o A198772 A198772list(upto) = [n for n in 0:upto if isA198772(n)]
%o A198772 A198772list(193) |> println # _Peter Luschny_, Mar 17 2018
%Y A198772 Subsequence of Loeschian numbers A003136.
%Y A198772 Complement of A118886 with respect to A003136.
%Y A198772 Cf. A198773, A198774, A198775.
%K A198772 nonn
%O A198772 1,3
%A A198772 _Reinhard Zumkeller_, Oct 30 2011