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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 (list; graph; refs; listen; history; text; internal format)
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 = (x-1)/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. 104-134, Dover Publ., NY, 1964.

Andrew Granville, Solution to Problem 90:07, Western Number Theory Problems, 1991-12-19 & 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), 31-42.

FORMULA

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

EXAMPLE

6 is in the sequence because it is the area of the 3-4-5 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

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Last modified November 17 21:02 EST 2018. Contains 317278 sequences. (Running on oeis4.)