A156681 Consider all Pythagorean triangles A^2 + B^2 = C^2 with A < B < C; sequence gives values of B, sorted to correspond to increasing A (A009004).

4, 12, 8, 24, 15, 12, 40, 24, 60, 16, 35, 84, 48, 20, 36, 112, 30, 63, 144, 24, 80, 180, 21, 48, 99, 28, 72, 220, 120, 264, 32, 45, 70, 143, 60, 312, 168, 36, 120, 364, 45, 96, 195, 420, 40, 72, 224, 480, 60, 126, 255, 44, 56, 180, 544, 288, 84, 120, 612, 48, 77, 105

Offset

1,1

Comments

The ordered sequence of B values is A009012(n) (allowing repetitions) and A009023(n) (excluding repetitions).

References

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

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

Links

Shujing Lyu, Table of n, a(n) for n = 1..5000

Ron Knott, Right-angled Triangles and Pythagoras' Theorem

Robert Recorde, The Whetstone of Witte, whiche is the seconde parte of Arithmeteke: containing the extraction of rootes; the cossike practise, with the rule of equation; and the workes of Surde Nombers, London, 1557. See p. 57.

Formula

a(n) = sqrt(A156682(n)^2 - A009004(n)^2).

Example

As the first four Pythagorean triples (ordered by increasing A) are (3,4,5), (5,12,13), (6,8,10) and (7,24,25), then a(1)=4, a(2)=12, a(3)=8 and a(4)=24.

Mathematica

PythagoreanTriplets[n_]:=Module[{t={{3, 4, 5}}, i=4, j=5}, While[i<n, h=Sqrt[i^2+j^2]; If[IntegerQ[h] && j<n, AppendTo[t, {i, j, h}]]; If[j<n, j++, i++; j=i+1]]; t]; k=20; data1=PythagoreanTriplets[2k^2+2k+1]; data2=Select[data1, #[[1]]<=2k+1 &]; #[[2]] &/@data2

Crossrefs

Cf. A156682, A009004, A009012, A009023.

Sequence in context: A208855 A252984 A084415 * A231100 A229179 A273172

Adjacent sequences: A156678 A156679 A156680 * A156682 A156683 A156684

Keyword

easy,nice,nonn

Author

Ant King, Feb 17 2009

Status

approved