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%I #6 Oct 23 2017 06:12:51
%S 1,7,17,7,31,49,23,71,97,47,127,161,1,79,199,241,119,287,337,17,167,
%T 391,449,223,23,511,577,41,287,647,41,721,359,799,881,73,439,967,1057,
%U 7,527,1151,89,1249,113,623,1351,1457,727,119,1567,1681,31,161,839,1799,1921
%N Consider primitive Pythagorean triangles (A^2 + B^2 = C^2, gcd (A, B) = 1, A < B<C); sequence gives values of B-A, sorted to correspond to increasing A (A020884(n)).
%C This sequence contains the differences in the legs of the primitive Pythagorean triples, sorted by shortest side (A020884). If a difference appears once then it must appear infinitely often, for if (m,n) generates a primitive triple with Abs(b-a)=d then so too does (2m+n,m). This corresponds to applying Hall's A matrix, and hence all horizontal lines in the Pythagorean family tree will contain families of primitive triples whose legs differ by the same amount. The sorted differences that can occur are in A058529.
%D Barning, F. J. M.; On Pythagorean and quasi-Pythagorean triangles and a generation process with the help of unimodular matrices. (Dutch), Math. Centrum Amsterdam Afd. Zuivere Wisk. ZW-001 (1963).
%H A. Hall, <a href="http://www.jstor.org/stable/3613860">Genealogy of Pythagorean Triads</a>, The Mathematical Gazette, Vol. 54, No. 390, (December 1970), pp. 377-379.
%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>
%F a(n) = A156678(n) - A020884(n).
%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)=4-3=1, a(2)=12-5=7, a(3)=24-7=17 and a(4)=15-8=7.
%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 &];#[[2]]-#[[1]] &/@data2
%Y Cf. A020884, A155678, A058529.
%K easy,nice,nonn
%O 1,2
%A _Ant King_, Feb 15 2009
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