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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 STATUS approved

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Last modified December 15 22:02 EST 2019. Contains 330012 sequences. (Running on oeis4.)