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A115880
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Largest positive x satisfying the Diophantine equation x^2 = y*(y+n), a(n)=0 if there are no solutions.
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4
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0, 0, 2, 0, 6, 4, 12, 3, 20, 12, 30, 8, 42, 24, 56, 15, 72, 40, 90, 24, 110, 60, 132, 35, 156, 84, 182, 48, 210, 112, 240, 63, 272, 144, 306, 80, 342, 180, 380, 99, 420, 220, 462, 120, 506, 264, 552, 143, 600, 312, 650, 168, 702, 364, 756, 195, 812, 420, 870
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OFFSET
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1,3
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COMMENTS
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Notice that x^2 = y*(y+n) is equivalent to (n+2*y+2*x)*(n+2*y-2*x) = n^2. We take the factorization of n^2 into two factors congruent mod 4 where one is as small as possible and the other is as large as possible. For n == 0 mod 4 the factors are 4 and n^2/4, for n == 2 mod 4 they are 2 and n^2/2, for n odd they are 1 and n^2. - Robert Israel, Jun 27 2014
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LINKS
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FORMULA
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Empirical g.f.: x^3*(x^9-2*x^6-3*x^5-6*x^4-4*x^3-6*x^2-2) / ((x-1)^3*(x+1)^3*(x^2+1)^3). - Colin Barker, Jun 26 2014
a(4*j) = j^2 - 1,
a(4*j+1) = 4*j^2+2*j,
a(4*j+2) = 2*j^2+2*j,
a(4*j+3) = 4*j^2+6*j+2. (see Comments) - Robert Israel, Jun 27 2014
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EXAMPLE
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a(15)=56 since the solutions (x,y) to x^2=y(y+15) are (4,1), (10,5), (18, 12) and (56, 49). The largest x is 56, from (x,y)=(56,49).
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MATHEMATICA
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Table[Max[x/.Solve[{x^2==y(y+n), x>0}, {x, y}, Integers]], {n, 1, 100}]/.x->0 (* Vaclav Kotesovec, Jun 26 2014 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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