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 A156679 Consider primitive Pythagorean triangles (A^2 + B^2 = C^2, gcd (A, B) = 1, A < B

%I

%S 5,13,25,17,41,61,37,85,113,65,145,181,29,101,221,265,145,313,365,53,

%T 197,421,481,257,65,545,613,85,325,685,89,761,401,841,925,125,485,

%U 1013,1105,73,577,1201,149,1301,173,677,1405,1513,785,185,1625,1741,109,229

%N Consider primitive Pythagorean triangles (A^2 + B^2 = C^2, gcd (A, B) = 1, A < B<C); sequence gives values of C, sorted to correspond to increasing A (A020884(n)).

%C The ordered sequence of A values is A020884(n) and the ordered sequence of C values is A020882(n) (allowing repetitions) and A008846(n) (excluding repetitions).

%D Beiler, Albert H.: Recreations In The Theory Of Numbers, Chapter XIV, The Eternal Triangle, Dover Publications Inc., New York, 1964, pp. 104-134.

%D Sierpinski, W.; Pythagorean Triangles, Dover Publications, Inc., Mineola, New York, 2003.

%H Reinhard Zumkeller, <a href="/A156679/b156679.txt">Table of n, a(n) for n = 1..1000</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>

%e 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)=5, a(2)=13, a(3)=25 and a(4)=17.

%t 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 &];#[[3]] &/@data2

%t With[{nn=61},Take[Sqrt[#[[1]]^2+#[[2]]^2]&/@Union[Sort/@({Times@@#,(Last[ #]^2- First[#]^2)/2}&/@(Select[Subsets[Range[1,nn,2],{2}],GCD@@# == 1&]))],nn]] (* _Harvey P. Dale_, May 10 2020 *)

%o a156679 n = a156679_list !! (n-1)

%o a156679_list = f 1 1 where

%o f u v | v > uu `div` 2 = f (u + 1) (u + 2)

%o | gcd u v > 1 || w == 0 = f u (v + 2)

%o | otherwise = w : f u (v + 2)

%o where uu = u ^ 2; w = a037213 (uu + v ^ 2)

%o -- _Reinhard Zumkeller_, Nov 09 2012

%Y Cf. A020884, A020882, A008846, A156678, A156682.

%Y Cf. A037213.

%K easy,nice,nonn

%O 1,1

%A _Ant King_, Feb 15 2009

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Last modified May 9 19:47 EDT 2021. Contains 343746 sequences. (Running on oeis4.)