OFFSET
1,1
COMMENTS
The triangles need not be primitive. Number of terms less than 10^n: 0, 0, 1, 3, 14, 53, ....
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.
From Sture Sjöstedt, Dec 06 2016: (Start)
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.
10920 = 7*8*13*15; p=7, q=15, q-p=8, r=13.
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)
REFERENCES
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).
LINKS
Giovanni Resta, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pythagorean Triple.
FORMULA
EXAMPLE
a(1) = 840 is the area of {15,112,113}, {24,70,74} & {40,42,58}.
a(2) = 3360 is the area of {30,224,226}, {48,140,148} & {80,84,116}.
a(3) = 7560 is the area of {45,336,339}, {72,210,222} & {120,126,174}.
MATHEMATICA
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 &]
CROSSREFS
KEYWORD
nonn
AUTHOR
Claudio Meller, on a suggestion by Antonio Roldán, Dec 08 2010
EXTENSIONS
Extended and edited by Robert G. Wilson v, Dec 08 2010
a(28)-a(34) from Giovanni Resta, Aug 16 2017
STATUS
approved