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

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

Cf. A007237, A290451.

Sequence in context: A011931 A260981 A078866 * A342320 A341759 A144620

Adjacent sequences: A332686 A332687 A332688 * A332690 A332691 A332692

Keyword

nonn

Author

Jeppe Stig Nielsen, Feb 19 2020

Extensions

a(8)-a(54) from Giovanni Resta, Mar 04 2020

Status

approved