A333917 Perimeters of integer-sided triangles whose altitude from their longest side is an integer.

16, 18, 32, 36, 48, 50, 54, 60, 64, 70, 72, 80, 90, 96, 98, 100, 108, 112, 120, 126, 128, 132, 140, 144, 150, 154, 160, 162, 168, 176, 180, 192, 196, 198, 200, 208, 210, 216, 220, 224, 234, 240, 242, 250, 252, 256, 260, 264, 270, 272, 280, 286, 288, 290, 294, 300

Offset

1,1

Links

Table of n, a(n) for n=1..56.

Eric Weisstein's World of Mathematics, Altitude

Wikipedia, Altitude (triangle)

Wikipedia, Integer Triangle

Example

16 is in the sequence since it is the perimeter of the triangle [5,5,6], whose altitude from 6 (the longest side) is 4 (an integer).
18 is in the sequence since it is the perimeter of the triangle [5,5,8], whose altitude from 8 (the longest side) is 3 (an integer).
48 is in the sequence since it is the perimeter of the triangles [15,15,18] and [10,17,21], whose altitudes from their longest sides are 12 and 8 respectively (both integers).

Mathematica

Flatten[Table[If[Sum[Sum[(1 - Ceiling[2*Sqrt[(n/2) (n/2 - i) (n/2 - k) (n/2 - (n - i - k))]/(n - i - k)] + Floor[2*Sqrt[(n/2) (n/2 - i) (n/2 - k) (n/2 - (n - i - k))]/(n - i - k)]) Sign[Floor[(i + k)/(n - i - k + 1)]], {i, k, Floor[(n - k)/2]}], {k, Floor[n/3]}] > 0, n, {}], {n, 100}]]

Crossrefs

Cf. A005044, A333918, A333919.

Sequence in context: A334453 A335243 A334586 * A335248 A308222 A278471

Adjacent sequences: A333914 A333915 A333916 * A333918 A333919 A333920

Keyword

nonn

Author

Wesley Ivan Hurt, Apr 09 2020

Status

approved