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A115881
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a(n) is the largest positive y 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, 1, 0, 4, 2, 9, 1, 16, 8, 25, 4, 36, 18, 49, 9, 64, 32, 81, 16, 100, 50, 121, 25, 144, 72, 169, 36, 196, 98, 225, 49, 256, 128, 289, 64, 324, 162, 361, 81, 400, 200, 441, 100, 484, 242, 529, 121, 576, 288, 625, 144, 676, 338, 729, 169, 784, 392, 841, 196
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OFFSET
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1,5
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COMMENTS
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The corresponding least y is given by A067721(n).
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LINKS
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FORMULA
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Empirical g.f.: -x^3*(x^9+x^8+2*x^7+4*x^6+x^5+6*x^4+2*x^3+4*x^2+1) / ((x-1)^3*(x+1)^3*(x^2+1)^3). - Colin Barker, Jun 26 2014
From empirical g.f.: a(n) = 1/2 - n/2 + 11*n^2/64 + (1/4 - 1/32*n^2) * (2*floor(n/4) + 2*floor((n+1)/4) - n + 1) + (1/4 - 5/64*n^2)*(-1)^n. - Vaclav Kotesovec, Jun 26 2014
a(4*j) = j^2 - 2*j + 1,
a(4*j+1) = 4*j^2,
a(4*j+2) = 2*j^2,
a(4*j+3) = 4*j^2+4*j+1 (see A115880).
(End)
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EXAMPLE
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a(15)=49, since the solutions (x,y) to x^2=y(y+15) are (4,1), (10,5), (18, 12) and (56, 49). The largest y is 49, from (x,y)=(56,49).
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MATHEMATICA
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Table[Max[y/.Solve[{x^2==y*(y+n), y>0}, {x, y}, Integers]], {n, 1, 100}]/.y->0 (* Vaclav Kotesovec, Jun 26 2014 *)
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PROG
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(Python)
a, b = divmod(n, 4)
return ((c:=a**2)-(a<<1)+1, (d:=c<<2), c<<1, d+(a<<2)+1)[b] # Chai Wah Wu, Aug 21 2024
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CROSSREFS
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KEYWORD
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nonn,changed
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AUTHOR
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STATUS
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approved
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