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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. 1
1, -1, -2, 3, 2, -5, -3, 8, -4, -12, -8, 4, 12, -16, -28, 44, 24, -20, -44, -24, 20, 56, 32, -88, 48, 40, -112, 64, 176, -48, -128, -64, 192, 256, -256, -512, 768, 512, -1280, -768, 2048, -1024, -3072, -2048, 1024, 3072, -4096, -7168, 11264, 6144, -5120, -11264, -6144, 5120, 14336, 8192 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
COMMENTS
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.
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.
LINKS
EXAMPLE
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.
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.
CROSSREFS
Sequence in context: A265675 A094020 A165609 * A141465 A263216 A141663
KEYWORD
sign
AUTHOR
Scott R. Shannon, Nov 17 2022
STATUS
approved

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Last modified June 18 11:45 EDT 2024. Contains 373481 sequences. (Running on oeis4.)