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A156680 Consider primitive Pythagorean triangles (A^2 + B^2 = C^2, gcd (A, B) = 1, A < B sequence gives values of B-A, sorted to correspond to increasing A (A020884(n)).

%I A156680 #10 Jul 25 2024 02:42:15
%S A156680 1,7,17,7,31,49,23,71,97,47,127,161,1,79,199,241,119,287,337,17,167,
%T A156680 391,449,223,23,511,577,41,287,647,41,721,359,799,881,73,439,967,1057,
%U A156680 7,527,1151,89,1249,113,623,1351,1457,727,119,1567,1681,31,161,839,1799,1921
%N A156680 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 A156680 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 A156680 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 A156680 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 A156680 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 A156680 a(n) = A156678(n) - A020884(n).
%e A156680 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 A156680 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 A156680 Cf. A020884, A156678, A058529.
%K A156680 easy,nice,nonn
%O A156680 1,2
%A A156680 _Ant King_, Feb 15 2009