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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(n)). 6
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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

REFERENCES

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

Sierpinski, W.; 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

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

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Last modified November 17 19:40 EST 2017. Contains 294834 sequences.