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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)). 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 (list; graph; refs; listen; history; text; internal format)
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(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.

REFERENCES

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).

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. 377-379.

Ron Knott, Right-angled 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)=4-3=1, a(2)=12-5=7, a(3)=24-7=17 and a(4)=15-8=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

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Last modified April 9 14:04 EDT 2020. Contains 333353 sequences. (Running on oeis4.)