A007237 Number of triangles with integer sides and area = n times perimeter.

5, 18, 45, 45, 52, 139, 80, 89, 184, 145, 103, 312, 96, 225, 379, 169, 116, 498, 123, 328, 560, 280, 134, 592, 228, 271, 452, 510, 134, 1036, 144, 280, 639, 339, 597, 1119, 139, 354, 635, 648, 162, 1486, 169, 594, 1215, 354, 186, 1066, 369, 622, 706, 597, 164

Offset: 1

Author

Simon Plouffe

Keywords

• nonn

Examples

			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).
		

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Ray Chandler, Table of n, a(n) for n = 1..5000 (first 300 terms from Zhining Yang)
• Lubomir Markov, Heronian Triangles Whose Areas Are Integer Multiples of Their Perimeters, Forum Geometricorum, Volume 7 (2007), 129-135.
• Juan V. Savall and Jesus Ferrer, Problem E3408, Amer. Math. Monthly, 99 (1992), 175-176.

Crossrefs

Cf. A120062.

Programs

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, ", "))

Python

from math import sqrt, floor
def A007237(n):
    ct = 0; k = 4*n*n
    for x in range(1, floor(2*sqrt(3)*n) + 1):
        for y in range(max(k//x + 1, x), floor((k+2*n*sqrt(k+x*x))/x)+1):
            if k*(x + y)%(x*y - k) == 0: ct += 1
    return ct  # Ya-Ping Lu, Dec 24 2023

Formula

a(n) = A120062(2n). - Ray Chandler, Jul 27 2017

Extensions

a(16)-a(50) from Les Reid, Jul 06 2010