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

Author

Wesley Ivan Hurt, Apr 09 2020

Keywords

• nonn

Links

• Eric Weisstein's World of Mathematics, Altitude
• Wikipedia, Altitude (triangle)
• Wikipedia, Integer Triangle

Crossrefs

Cf. A005044, A333917, A333919.

Programs

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}]]

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).