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A156689
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Inradii of primitive Pythagorean triples a^2+b^2=c^2, 0<a<b<c with gcd(a,b)=1, and sorted to correspond to increasing a (given in A020884).
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1
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1, 2, 3, 3, 4, 5, 5, 6, 7, 7, 8, 9, 6, 9, 10, 11, 11, 12, 13, 10, 13, 14, 15, 15, 12, 16, 17, 14, 17, 18, 15, 19, 19, 20, 21, 18, 21, 22, 23, 15, 23, 24, 21, 25, 22, 25, 26, 27, 27, 24, 28, 29, 21, 26, 29, 30
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OFFSET
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1,2
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COMMENTS
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The inradius is given by r=1/2 (a+b-c)=ab/(a+b+c)=area/semiperimeter, and the inradii ordered by increasing r are in A020888.
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REFERENCES
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Mohammad K. Azarian, Circumradius and Inradius, Problem S125, Math Horizons, Vol. 15, Issue 4, April 2008, p. 32. Solution published in Vol. 16, Issue 2, November 2008, p. 32.
D. G. Rogers, Putting Pythagoras in the frame, Mathematics Today, The Institute of Mathematics and its Applications, Vol. 44, No. 3, June 2008, pp. 123-125.
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LINKS
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Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Ron Knott, Right-angled Triangles and Pythagoras' Theorem
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FORMULA
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A156689(n)=1/2 (A020884(n)+A156678(n)-A156679(n))
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EXAMPLE
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The eighth primitive Pythagorean triple ordered by increasing a is (13,84,85). As this has inradius 1/2 (13+84-85)=6, we have a(8)=6.
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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=30; data1=PrimitivePythagoreanTriplets[2k^2+2k+1]; data2=Select[data1, #[[1]]<=2k+1 &]; 1/2(#[[1]]+#[[2]]-#[[3]]) &/@data2
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PROG
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(Haskell)
a156689 n = a156689_list !! (n-1)
a156689_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 = (u + v - w) `div` 2 : f u (v + 2)
where uu = u ^ 2; w = a037213 (uu + v ^ 2)
-- Reinhard Zumkeller, Nov 09 2012
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CROSSREFS
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Cf. A020884, A020888, A156678, A156679.
Cf. A037213.
Sequence in context: A189726 A093878 A317686 * A168052 A131737 A004396
Adjacent sequences: A156686 A156687 A156688 * A156690 A156691 A156692
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KEYWORD
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easy,nice,nonn
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AUTHOR
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Ant King, Feb 18 2009
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STATUS
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approved
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