

A156689


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).


1



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

1,2


COMMENTS

The inradius is given by r=1/2 (a+bc)=ab/(a+b+c)=area/semiperimeter, and the inradii ordered by increasing r are in A020888.


REFERENCES

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. 123125.


LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Ron Knott, Rightangled Triangles and Pythagoras' Theorem


FORMULA

A156689(n)=1/2 (A020884(n)+A156678(n)A156679(n))


EXAMPLE

The eighth primitive Pythagorean triple ordered by increasing a is (13,84,85). As this has inradius 1/2 (13+8485)=6, we have a(8)=6.


MATHEMATICA

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


PROG

(Haskell)
a156689 n = a156689_list !! (n1)
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


CROSSREFS

Cf. A020884, A020888, A156678, A156679.
Cf. A037213.
Sequence in context: A189726 A093878 A317686 * A168052 A131737 A004396
Adjacent sequences: A156686 A156687 A156688 * A156690 A156691 A156692


KEYWORD

easy,nice,nonn


AUTHOR

Ant King, Feb 18 2009


STATUS

approved



