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A024363
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Number of primitive Pythagorean triangles with side n.
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7
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0, 0, 1, 1, 2, 0, 1, 1, 1, 0, 1, 2, 2, 0, 2, 1, 2, 0, 1, 2, 2, 0, 1, 2, 2, 0, 1, 2, 2, 0, 1, 1, 2, 0, 2, 2, 2, 0, 2, 2, 2, 0, 1, 2, 2, 0, 1, 2, 1, 0, 2, 2, 2, 0, 2, 2, 2, 0, 1, 4, 2, 0, 2, 1, 4, 0, 1, 2, 2, 0, 1, 2, 2, 0, 2, 2, 2, 0, 1, 2, 1, 0, 1, 4, 4, 0, 2, 2, 2, 0, 2, 2, 2, 0, 2, 2, 2, 0, 2
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OFFSET
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1,5
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COMMENTS
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Consider primitive Pythagorean triangles (A^2 + B^2 = C^2, (A, B) = 1, A <= B); sequence gives number of times AUBUC takes value n.
Using Euclidean parameters (x, y) with x > y to generate primitive Pythagorean triples to capture all occurrences of side n, the Mma program below must allow the x parameter to iterate at least (n+1)/2 times. - Frank M Jackson, Jun 12 2017
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LINKS
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FORMULA
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MATHEMATICA
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lst={}; xmax=51; Do[If[GCD[x, y]==1&&OddQ[x+y], AppendTo[lst, Sort@{x^2-y^2, 2 x*y, x^2+y^2}]], {x, xmax}, {y, x}]; BinCounts[Select[Flatten@lst, #<2xmax &], {1, 2(xmax-1), 1}] (* or *)
a[n_] := Block[{x, y, s = List@ ToRules@ Reduce[(x^2-y^2 == n^2 || x^2 + y^2 == n^2) && x>y>0, {x, y}, Integers]}, If[s == {}, 0, Length@ Select[ {x, y} /. s, GCD @@ # == 1 &]]]; Array[a, 99] (* Giovanni Resta, Jun 19 2017 *)
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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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