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A156678
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Consider primitive Pythagorean triangles (A^2 + B^2 = C^2, gcd (A, B) = 1, A < B<C); sequence gives values of B, sorted to correspond to increasing A (A020884(n)).
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7
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4, 12, 24, 15, 40, 60, 35, 84, 112, 63, 144, 180, 21, 99, 220, 264, 143, 312, 364, 45, 195, 420, 480, 255, 56, 544, 612, 77, 323, 684, 80, 760, 399, 840, 924, 117, 483, 1012, 1104, 55, 575, 1200, 140, 1300, 165, 675, 1404, 1512, 783, 176, 1624, 1740, 91, 221, 899
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OFFSET
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1,1
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COMMENTS
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The ordered sequence of A values is A020884(n) and the ordered sequence of B values is A020883(n) (allowing repetitions) and A024354(n) (excluding repetitions)
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REFERENCES
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Beiler, Albert H.: Recreations In The Theory Of Numbers, Chapter XIV, The Eternal Triangle, Dover Publications Inc., New York, 1964, pp. 104-134.
Sierpinski, W.; Pythagorean Triangles, Dover Publications, Inc., Mineola, New York, 2003.
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LINKS
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FORMULA
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EXAMPLE
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As the first four primitive Pythagorean triples (ordered by increasing A) are (3,4,5), (5,12,13), (7,24,25) and (8,15,17), then a(1)=4, a(2)=12, a(3)=24 and a(4)=15.
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MATHEMATICA
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PrimitivePythagoreanTriplets[n_]:=Module[{t={{3, 4, 5}}, i=4, j=5}, While[i<n, If[GCD[i, j]==1, h=Sqrt[i^2+j^2]; If[IntegerQ[h] && j<n, AppendTo[t, {i, j, h}]]; ]; If[j<n, j+=2, i++; j=i+1]]; t]; k=38; data1=PrimitivePythagoreanTriplets[2k^2+2k+1]; data2=Select[data1, #[[1]]<=2k+1 &]; #[[2]] &/@data2
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PROG
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(Haskell)
a156678 n = a156678_list !! (n-1)
a156678_list = f 1 1 where
f u v | v > uu `div` 2 = f (u + 1) (u + 2)
| gcd u v > 1 || w == 0 = f u (v + 2)
| otherwise = v : f u (v + 2)
where uu = u ^ 2; w = a037213 (uu + v ^ 2)
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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