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%I #17 Jan 03 2022 03:07:01
%S 4,12,8,24,15,12,40,24,60,16,35,84,48,20,36,112,30,63,144,24,80,180,
%T 21,48,99,28,72,220,120,264,32,45,70,143,60,312,168,36,120,364,45,96,
%U 195,420,40,72,224,480,60,126,255,44,56,180,544,288,84,120,612,48,77,105
%N Consider all Pythagorean triangles A^2 + B^2 = C^2 with A < B < C; sequence gives values of B, sorted to correspond to increasing A (A009004).
%C The ordered sequence of B values is A009012(n) (allowing repetitions) and A009023(n) (excluding repetitions).
%D Albert H. Beiler, Recreations In The Theory Of Numbers, Chapter XIV, The Eternal Triangle, Dover Publications Inc., New York, 1964, pp. 104-134.
%D W. Sierpinski, Pythagorean Triangles, Dover Publications, Inc., Mineola, New York, 2003.
%H Shujing Lyu, <a href="/A156681/b156681.txt">Table of n, a(n) for n = 1..5000</a>
%H Ron Knott, <a href="http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Pythag/pythag.html">Right-angled Triangles and Pythagoras' Theorem</a>
%H Robert Recorde, <a href="http://archive.org/details/TheWhetstoneOfWitte">The Whetstone of Witte</a>, whiche is the seconde parte of Arithmeteke: containing the extraction of rootes; the cossike practise, with the rule of equation; and the workes of Surde Nombers, London, 1557. See <a href="http://archive.org/stream/TheWhetstoneOfWitte#page/n55/mode/2up/">p. 57</a>.
%F a(n) = sqrt(A156682(n)^2 - A009004(n)^2).
%e As the first four Pythagorean triples (ordered by increasing A) are (3,4,5), (5,12,13), (6,8,10) and (7,24,25), then a(1)=4, a(2)=12, a(3)=8 and a(4)=24.
%t PythagoreanTriplets[n_]:=Module[{t={{3,4,5}},i=4,j=5},While[i<n,h=Sqrt[i^2+j^2];If[IntegerQ[h] && j<n,AppendTo[t,{i,j,h}]]; If[j<n,j++,i++;j=i+1]];t];k=20;data1=PythagoreanTriplets[2k^2+2k+1];data2=Select[data1,#[[1]]<=2k+1 &];#[[2]] &/@data2
%Y Cf. A156682, A009004, A009012, A009023.
%K easy,nice,nonn
%O 1,1
%A _Ant King_, Feb 17 2009
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