A333919 Perimeters of integer-sided triangles with side lengths a <= b <= c whose altitude from side b is an integer.

12, 24, 30, 36, 40, 42, 48, 56, 60, 70, 72, 78, 80, 84, 90, 96, 104, 108, 110, 112, 114, 120, 126, 132, 136, 140, 144, 150, 154, 156, 160, 162, 168, 176, 180, 182, 186, 192, 198, 200, 204, 208, 210, 216, 220, 222, 224, 228, 230, 232, 234, 238, 240, 250, 252

Offset

1,1

Links

Table of n, a(n) for n=1..55.

Eric Weisstein's World of Mathematics, Altitude

Wikipedia, Altitude (triangle)

Wikipedia, Integer Triangle

Example

12 is in the sequence since it is the perimeter of the triangle [3,4,5], whose altitude from 4 (its "middle" side) is 3 (an integer).
24 is in the sequence since it is the perimeter of the triangle [6,8,10], whose altitude from 8 (its "middle" side) is 6 (an integer).
60 is in the sequence since it is the perimeter of the triangles [10,24,26] and [15,20,25], whose altitudes (from their "middle" sides) are 10 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))]/i] + Floor[2*Sqrt[(n/2) (n/2 - i) (n/2 - k) (n/2 - (n - i - k))]/i]) 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, A333918.

Sequence in context: A365008 A182225 A074697 * A289484 A027854 A009096

Adjacent sequences: A333916 A333917 A333918 * A333920 A333921 A333922

Keyword

nonn

Author

Wesley Ivan Hurt, Apr 09 2020

Status

approved