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%I M3878 #48 Dec 27 2023 09:02:31
%S 5,18,45,45,52,139,80,89,184,145,103,312,96,225,379,169,116,498,123,
%T 328,560,280,134,592,228,271,452,510,134,1036,144,280,639,339,597,
%U 1119,139,354,635,648,162,1486,169,594,1215,354,186,1066,369,622,706,597,164
%N Number of triangles with integer sides and area = n times perimeter.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Ray Chandler, <a href="/A007237/b007237.txt">Table of n, a(n) for n = 1..5000</a> (first 300 terms from Zhining Yang)
%H Lubomir Markov, <a href="https://forumgeom.fau.edu/FG2007volume7/FG200718index.html">Heronian Triangles Whose Areas Are Integer Multiples of Their Perimeters</a>, Forum Geometricorum, Volume 7 (2007), 129-135.
%H Juan V. Savall and Jesus Ferrer, <a href="http://www.jstor.org/stable/2324200">Problem E3408</a>, Amer. Math. Monthly, 99 (1992), 175-176.
%F a(n) = A120062(2n). - _Ray Chandler_, Jul 27 2017
%e For n=2, the a(2)=18 solutions whose area is twice its perimeter: (13,14,15) (12,16,20) (15,15,24) (10,24,26) (11,25,30) (18,20,34) (15,26,37) (14,30,40) (10,35,39) (9,40,41) (12,50,58) (33,34,65) (25,51,74) (9,75,78) (11,90,97) (21,85,104) (19,153,170) (18,289,305).
%o (PARI) for(k=1,100, n=0; d=4*k^2; e=3*d; for(b=1,sqrt(e), for(c=2*k,e/b, if(b*c>d && c>=b, f = (b + c)*d / (b * c - d); if(f>=c, a=floor(f); if(a==f, n++))))); print1(n, ", "))
%o (Python)
%o from math import sqrt, floor
%o def A007237(n):
%o ct = 0; k = 4*n*n
%o for x in range(1, floor(2*sqrt(3)*n) + 1):
%o for y in range(max(k//x + 1, x), floor((k+2*n*sqrt(k+x*x))/x)+1):
%o if k*(x + y)%(x*y - k) == 0: ct += 1
%o return ct # _Ya-Ping Lu_, Dec 24 2023
%Y Cf. A120062.
%K nonn
%O 1,1
%A _Simon Plouffe_
%E a(16)-a(50) from _Les Reid_, Jul 06 2010
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