

A009111


List of ordered areas of Pythagorean triangles.


15



6, 24, 30, 54, 60, 84, 96, 120, 150, 180, 210, 210, 216, 240, 270, 294, 330, 336, 384, 480, 486, 504, 540, 546, 600, 630, 720, 726, 750, 756, 840, 840, 840, 864, 924, 960, 990, 1014, 1080, 1176, 1224, 1320, 1320, 1344, 1350, 1386, 1470, 1500, 1536, 1560, 1620
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OFFSET

1,1


COMMENTS

All terms are divisible by 6.
Let k be even, k > 2, q = (k/2)^2  1, and b = (kq)/2. Then, for any k, b is a term of a(n). In other words, for any even k > 2, there is at least one such integer q > 2 that b = (kq)/2 and b is a term of a(n), while hypotenuse c = q + 2 (proved by Anton Mosunov).  Sergey Pavlov, Mar 02 2017
Let x be odd, x > 1, k == 0 (mod x), k > 0, y = (x1)/2, q = ky + (ky/x), b = (kq)/2. Then b is a term of a(n), while hypotenuse c = q + k/x. As a special case of the above equation (k = x), for each odd k > 1 there exist such q and b that q = (k^2  1)/2, b = (kq)/2, and b is a term of a(n), while hypotenuse c = q + 1.  Sergey Pavlov, Mar 06 2017


REFERENCES

Albert H. Beiler, Recreations in the Theory of Numbers, The Queen of Mathematics Entertains, 2nd Ed., Chpt. XIV, "The Eternal Triangle", pp. 104134, Dover Publ., NY, 1964.
Andrew Granville, Solution to Problem 90:07, Western Number Theory Problems, 19911219 & 22, ed. R. K. Guy.


LINKS

T. D. Noe, Table of n, a(n) for n = 1..1000
Ron Knott, Pythagorean Triangles
Supriya Mohanty and S. P. Mohanty, Pythagorean Numbers, Fibonacci Quarterly 28 (1990), 3142.


FORMULA

Theorem: The number of pairs of integers a > b > 0 with ab(a^2b^2) < n^2 is Cn + O(n^(2/3)) where C = (1/2)*Integral_{1..infinity} du/sqrt(u^3u). [Granville]  N. J. A. Sloane, Feb 07 2008


EXAMPLE

6 is in the sequence because it is the area of the 345 triangle.


MATHEMATICA

t = {}; nn = 200; mx = Sqrt[2*nn  1] (nn  1)/2; Do[x = Sqrt[n^2  d^2]; If[x > 0 && IntegerQ[x] && x > d && d*x/2 <= mx, AppendTo[t, d*x/2]], {n, nn}, {d, n  1}]; t = Sort[t]; t (* T. D. Noe, Sep 23 2013 *)


CROSSREFS

Cf. A009112, A024365.
Sequence in context: A131906 A185210 A046131 * A009112 A057101 A057228
Adjacent sequences: A009108 A009109 A009110 * A009112 A009113 A009114


KEYWORD

nonn


AUTHOR

David W. Wilson


STATUS

approved



