This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A177021 #45 Feb 16 2025 08:33:12
%S A177021 840,3360,7560,10920,13440,21000,30240,31920,41160,43680,53760,68040,
%T A177021 84000,98280,101640,120960,127680,141960,164640,166320,174720,189000,
%U A177021 215040,242760,272160,273000,286440,287280,303240,336000,370440,393120,406560,444360
%N A177021 Numbers which are the area of exactly three Pythagorean triangles.
%C A177021 The triangles need not be primitive. Number of terms less than 10^n: 0, 0, 1, 3, 14, 53, ....
%C A177021 13123110 is the smallest number which is the area of three primitive Pythagorean triangles, (1380,19019,19069)(3059,8580,9109) and (4485,5852,7373); this triple was found by Charles L. Shedd in 1945.
%C A177021 From _Sture Sjöstedt_, Dec 06 2016: (Start)
%C A177021 840 = 3*5*7*8; p=3, q=8, q-p=5, r=7 is a solution to p^2 - pq + q^2 = r^2. If r is a prime number in the sequence 7, 13, 19, ..., there are three Pythagorean triangles with the same area and at least one of them is primitive.
%C A177021 10920 = 7*8*13*15; p=7, q=15, q-p=8, r=13.
%C A177021 x^2 + 3*y^2 = 4*r^2 where r is a prime number in the sequence 7, 13, 19, ... gives lattice points that can be used to find solutions to p^2 - pq + q^2 = r^2. p, q, (q-p) and r are the y-coordinates in the first quadrant. (End)
%D A177021 Morton Cohen, Charles Lutwidge Dodgson (Lewis Carroll), b. Jan. 27, 1832, d. Jan. 14, 1898, A Brief Biography, Vintage Books, ISBN 978-0-679-74562-4 (26 November 1996).
%H A177021 Giovanni Resta, <a href="/A177021/b177021.txt">Table of n, a(n) for n = 1..10000</a>
%H A177021 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PythagoreanTriple.html">Pythagorean Triple</a>.
%H A177021 <a href="/index/Ps#PyTrip">Index entries related to Pythagorean Triples</a>.
%F A177021 A177021 = { n | A177063(n)=3 }. - _M. F. Hasler_, Dec 09 2010
%e A177021 a(1) = 840 is the area of {15,112,113}, {24,70,74} & {40,42,58}.
%e A177021 a(2) = 3360 is the area of {30,224,226}, {48,140,148} & {80,84,116}.
%e A177021 a(3) = 7560 is the area of {45,336,339}, {72,210,222} & {120,126,174}.
%t A177021 lst = {}; m = 2; While[ m < 10^3, n = 1; While[ n < m, If[ IntegerQ@ Sqrt[ m^2 + n^2], a = m*n/2; If[a < 10^6, AppendTo[ lst, a], n = m]]; n++ ]; m++ ]; Union@ Flatten@ Select[ Split@ Sort@ lst, Length@ # == 3 &]
%Y A177021 Cf. A009112, A177063.
%K A177021 nonn
%O A177021 1,1
%A A177021 _Claudio Meller_, on a suggestion by _Antonio Roldán_, Dec 08 2010
%E A177021 Extended and edited by _Robert G. Wilson v_, Dec 08 2010
%E A177021 a(28)-a(34) from _Giovanni Resta_, Aug 16 2017