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A334176
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Squarefree part of numerator of the squared area of the Heronian triangle with sequential odd sides whose shortest leg is 2*n+1.
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2
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3, 11, 195, 35, 51, 627, 91, 115, 51, 19, 203, 2139, 11, 35, 3219, 403, 451, 4515, 555, 611, 123, 731, 795, 7755, 19, 1003, 9699, 1155, 1235, 11859, 1403, 1491, 14235, 67, 1771, 16827, 219, 83, 19635, 2291, 267, 22659, 2635, 2755, 25899, 3003, 3131, 29355, 3395, 3531, 33027, 3811, 3955, 36915, 4251
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OFFSET
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1,1
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COMMENTS
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These are the values within the irreducible square roots, which are part of the area resulting from Heronian triangles, which have all sides being sequential odd integers (starting with the second odd number, since the triangle {1,3,5} has no real area). The triangles follow this sequence of sides: {2*n+1, 2*n+3, 2*n+5} and their area is represented by {(r/s)*sqrt(t)}, where r, s, t are integers and a(n) is the number t.
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LINKS
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FORMULA
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EXAMPLE
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a(1) = 3 because the first possible Heronian triangle with its sequential odd integer sides is the triangle {3,5,7} and its respective area is {15*sqrt(3)/4}. a(2) = 11 since the second possible triangle is {5,7,9} which has area {21*sqrt(11)/4}. And so on.
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MATHEMATICA
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a[n_]:=Module[{y, z}, z=Area@SSSTriangle[2*n+1, 2*n+3, 2*n+5]; ({z}/.Coefficient[{z}/.Sqrt[_]->y, y][[1]]->1)[[1]]^2]
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PROG
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(PARI) a(n) = core((3*(2*n+7)*(2*n-1))); \\ Michel Marcus, Apr 18 2020
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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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