

A156680


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


2



1, 7, 17, 7, 31, 49, 23, 71, 97, 47, 127, 161, 1, 79, 199, 241, 119, 287, 337, 17, 167, 391, 449, 223, 23, 511, 577, 41, 287, 647, 41, 721, 359, 799, 881, 73, 439, 967, 1057, 7, 527, 1151, 89, 1249, 113, 623, 1351, 1457, 727, 119, 1567, 1681, 31, 161, 839, 1799, 1921
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OFFSET

1,2


COMMENTS

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(ba)=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.


REFERENCES

Barning, F. J. M.; On Pythagorean and quasiPythagorean triangles and a generation process with the help of unimodular matrices. (Dutch), Math. Centrum Amsterdam Afd. Zuivere Wisk. ZW001 (1963).


LINKS

Table of n, a(n) for n=1..57.
A. Hall, Genealogy of Pythagorean Triads, The Mathematical Gazette, Vol. 54, No. 390, (December 1970), pp. 377379.
Ron Knott, Rightangled Triangles and Pythagoras' Theorem


FORMULA

a(n) = A156678(n)  A020884(n).


EXAMPLE

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)=43=1, a(2)=125=7, a(3)=247=17 and a(4)=158=7.


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=38; data1=PrimitivePythagoreanTriplets[2k^2+2k+1]; data2=Select[data1, #[[1]]<=2k+1 &]; #[[2]]#[[1]] &/@data2


CROSSREFS

Cf. A020884, A155678, A058529.
Sequence in context: A094464 A224795 A138449 * A107804 A276809 A274916
Adjacent sequences: A156677 A156678 A156679 * A156681 A156682 A156683


KEYWORD

easy,nice,nonn


AUTHOR

Ant King, Feb 15 2009


STATUS

approved



