A156678 - OEIS

A156678

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

4, 12, 24, 15, 40, 60, 35, 84, 112, 63, 144, 180, 21, 99, 220, 264, 143, 312, 364, 45, 195, 420, 480, 255, 56, 544, 612, 77, 323, 684, 80, 760, 399, 840, 924, 117, 483, 1012, 1104, 55, 575, 1200, 140, 1300, 165, 675, 1404, 1512, 783, 176, 1624, 1740, 91, 221, 899

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

OFFSET

1,1

COMMENTS

The ordered sequence of A values is A020884(n) and the ordered sequence of B values is A020883(n) (allowing repetitions) and A024354(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

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Ron Knott, Right-angled Triangles and Pythagoras' Theorem

FORMULA

a(n) = A020884(n) + A156680(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, a(2)=12, a(3)=24 and a(4)=15.

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]] &/@data2

PROG

(Haskell)

a156678 n = a156678_list !! (n-1)

a156678_list = f 1 1 where

f u v | v > uu `div` 2 = f (u + 1) (u + 2)

| gcd u v > 1 || w == 0 = f u (v + 2)

| otherwise = v : f u (v + 2)

where uu = u ^ 2; w = a037213 (uu + v ^ 2)

-- Reinhard Zumkeller, Nov 09 2012

CROSSREFS

A020884, A020883, A024354, A156679, A042965, A156680, A156681, A042965.

Cf. A037213.