A332689 - OEIS

A332689

Number of distinct areas of integer-sided triangles whose area equals n times their perimeter.

5, 17, 41, 41, 47, 127, 77, 81, 171, 132, 99, 283, 94, 205, 349, 158, 115, 457, 122, 296, 530, 267, 134, 546, 219, 260, 428, 471, 130, 953, 144, 264, 613, 332, 557, 1031, 139, 346, 614, 600, 162, 1381, 169, 562, 1132, 348, 186, 1000, 363, 593, 688, 571, 164, 1123

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Gives the row lengths of the irregular array A290451.

LINKS

Table of n, a(n) for n=1..54.

James Grime and Brady Haran, Superhero Triangles, Numberphile video (2020).

EXAMPLE

For n = 2, there are 18 different (noncongruent) Heronian triangles whose area equals twice their perimeter, so A007237(2) = 18. However, two of those 18 triangles share the area 168. So there are only 17 distinct areas. Therefore, a(2) = 17.

MATHEMATICA

a[k_] := Block[{v={}, r, s, t}, Do[ If[r <= s && 4 k^2 < r s <= 12 k^2 && IntegerQ[t = 4 k^2 (r + s)/(r s - 4 k^2)] && t >= s, AppendTo[v, r + s + t]], {r, Floor[2 Sqrt[3] k]}, {s, Floor[4 k^2/r], Ceiling[12 k^2/r]}]; Length@ Union@ v]; Array[a, 20] (* Giovanni Resta, Mar 04 2020 *)

PROG

(Python)

from math import sqrt

def A332689(n):

L = []; k = 4*n*n

for x in range(1, int(2*sqrt(3)*n) + 1):

for y in range(max(int(k/x) + 1, x), int((k + 2*n*sqrt(k + x*x))/x) + 1):

if k*(x+y)%(x*y-k) == 0:

s = x + y + k*(x+y)//(x*y-k)

if s not in L: L.append(s)

return len(L) # Ya-Ping Lu, Dec 28 2023

CROSSREFS