A333918 Perimeters of integer-sided triangles whose altitude from their shortest side is an integer.

12, 24, 30, 32, 36, 40, 42, 44, 48, 50, 54, 56, 60, 64, 66, 68, 70, 72, 76, 80, 84, 88, 90, 96, 98, 100, 104, 108, 112, 120, 126, 128, 130, 132, 136, 140, 144, 150, 152, 154, 156, 160, 162, 164, 168, 170, 172, 174, 176, 180, 182, 186, 190, 192, 196, 198, 200

Offset

1,1

Links

Table of n, a(n) for n=1..57.

Eric Weisstein's World of Mathematics, Altitude

Wikipedia, Altitude (triangle)

Wikipedia, Integer Triangle

Formula

12 is in the sequence since it is the perimeter of the triangle [3,4,5], whose altitude from 3 (the shortest side) is 4 (an integer).

24 is in the sequence since it is the perimeter of the triangle [6,8,10], whose altitude from 6 (the shortest side) is 8 (an integer).

54 is in the sequence since it is the perimeter of the triangles [3,25,26] and [12,17,25] whose altitudes from their shortest sides are 24 and 15 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))]/k] + Floor[2*Sqrt[(n/2) (n/2 - i) (n/2 - k) (n/2 - (n - i - k))]/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, A333917, A333919.

Sequence in context: A361155 A108902 A126935 * A334665 A333945 A335146

Adjacent sequences: A333915 A333916 A333917 * A333919 A333920 A333921

Keyword

nonn

Author

Wesley Ivan Hurt, Apr 09 2020

Status

approved