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Positive numbers of the form k = x^2 - 3*y^2.
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%I #44 May 22 2023 11:37:17

%S 1,4,6,9,13,16,22,24,25,33,36,37,46,49,52,54,61,64,69,73,78,81,88,94,

%T 96,97,100,109,117,118,121,132,141,142,144,148,150,157,166,169,177,

%U 181,184,193,196,198,208,213,214,216,222,225,229,241,244,249,253,256

%N Positive numbers of the form k = x^2 - 3*y^2.

%C Equivalently, positive numbers of the form k = x^2 + 2xy - 2y^2. These are equivalent forms, of discriminant 12.

%C Also numbers representable as x^2 + 4*x*y + y^2 with 0 <= x <= y. - _Gheorghe Coserea_, Jul 29 2018 [The restriction 0 <= x <= y is not necessary. - _Klaus Purath_, Feb 05 2023]

%C From _Klaus Purath_, Feb 05 2023: (Start)

%C Also positive numbers of the form x^2 + 2*m*x*y + (m^2 - 3)*y^2. This includes all forms given above so far.

%C All terms are congruent to {0, 1, 4, 6, 9, 10} modulo 12.

%C The product of any two terms belongs to the sequence - (empirically secured up to a(k)*a(m) for 2 <= k, m <= 85). Thus it appears that this sequence is closed under multiplication. Perhaps someone can find a proof? (End)

%H Jean-François Alcover, <a href="/A084916/b084916.txt">Table of n, a(n) for n = 1..1000</a>

%H Will Jagy, <a href="/A243655/a243655.txt">C++ program Conway_Positive_All.cc to find all positive numbers represented by an indefinite binary quadratic form</a>

%H Will Jagy, <a href="/A243655/a243655_2.txt">Sample output from Conway_Positive_All.cc</a>

%H Will Jagy, <a href="/A243655/a243655_1.txt">C++ program Conway_Positive_Primitive.cc to find positive numbers primitively represented by an indefinite binary quadratic form</a>

%H Will Jagy, <a href="/A243655/a243655_3.txt">Sample output from Conway_Positive_Prim.cc</a>

%H N. J. A. Sloane et al., <a href="https://oeis.org/wiki/Binary_Quadratic_Forms_and_OEIS">Binary Quadratic Forms and OEIS</a> (Index to related sequences, programs, references)

%t Reap[For[n = 1, n < 300, n++, If[Reduce[n == x^2 - 3*y^2, {x, y}, Integers] =!= False, Sow[n]]]][[2, 1]] (* _Jean-François Alcover_, Dec 03 2013 *)

%Y Cf. A031363, A035251, A243655 (primitive representations).

%Y See A068228 for primes.

%K nonn

%O 1,2

%A _Roger Cuculière_, Jul 14 2003

%E More terms from _Reinhard Zumkeller_, Jul 17 2003