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A358462 a(1) = 1, a(2) = -1; for n > 2, a(n) is smallest magnitude nonzero integer which has not appeared such that the quadratic equation a(n-2)*x^2 + a(n-1)*x + a(n) = 0 has at least one integer root.

%I A358462 #14 Jan 08 2023 13:01:59
%S A358462 1,-1,-2,3,2,-5,-3,8,-4,-12,-8,4,12,-16,-28,44,24,-20,-44,-24,20,56,
%T A358462 32,-88,48,40,-112,64,176,-48,-128,-64,192,256,-256,-512,768,512,
%U A358462 -1280,-768,2048,-1024,-3072,-2048,1024,3072,-4096,-7168,11264,6144,-5120,-11264,-6144,5120,14336,8192
%N A358462 a(1) = 1, a(2) = -1; for n > 2, a(n) is smallest magnitude nonzero integer which has not appeared such that the quadratic equation a(n-2)*x^2 + a(n-1)*x + a(n) = 0 has at least one integer root.
%C A358462 As a(8) and a(9) are both even, all subsequent terms will be even. This is due to the discriminant having to equal a square, and with both a(n-2) and a(n-1) being even, a(n) must also be even.
%C A358462 Although only one root must be an integer, several terms result in two integers as roots. For example a(3) = -2, a(4) = 3, a(11) = -8, a(14) = -16, a(34) = 256 all produce two integer roots.
%H A358462 Scott R. Shannon, <a href="/A358462/b358462.txt">Table of n, a(n) for n = 1..125</a>.
%H A358462 Wikipedia, <a href="http://en.wikipedia.org/wiki/Quadratic_equation">Quadratic equation</a>
%e A358462 a(3) = -2 as a(1)*x^2 + a(2)*x + a(3) = x^2 - x - 2 which has the integer roots x = -1 and x = 2, and -2 has not previously appeared.
%e A358462 a(6) = -5 as a(4)*x^2 + a(5)*x + a(6) = 3*x^2 + 2*x - 5 which has the integer root x = 1, and -5 has not previously appeared.
%Y A358462 Cf. A000290, A002378, A348139, A001622, A000058.
%K A358462 sign
%O A358462 1,3
%A A358462 _Scott R. Shannon_, Nov 17 2022