A007237 Number of triangles with integer sides and area = n times perimeter. (Formerly M3878)

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

1,1

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.

Formula

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

Example

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).

Prog

(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

Crossrefs

Cf. A120062.

Sequence in context: A101105 A037140 A321049 * A327842 A000339 A270944

Adjacent sequences: A007234 A007235 A007236 * A007238 A007239 A007240

Keyword

nonn

Author

Simon Plouffe

Extensions

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

Status

approved