A337086 Number of integer-sided triangles with perimeter n where the harmonic mean of the side lengths is an integer.

0, 0, 1, 0, 0, 1, 0, 0, 2, 1, 0, 1, 1, 0, 1, 0, 0, 2, 0, 1, 1, 0, 0, 1, 0, 1, 2, 1, 0, 2, 1, 0, 1, 0, 1, 3, 1, 0, 2, 1, 0, 2, 0, 0, 4, 1, 1, 2, 0, 1, 1, 2, 1, 2, 1, 2, 1, 0, 0, 2, 1, 2, 3, 1, 1, 1, 0, 0, 1, 2, 1, 3, 1, 1, 1, 0, 1, 2, 0, 1, 2, 0, 0, 3, 3, 1, 1, 0, 2, 6, 3, 1, 2, 1, 0, 2

Offset

1,9

Links

Table of n, a(n) for n=1..96.

Wikipedia, Integer Triangle

Formula

a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} sign(floor((i+k)/(n-i-k+1))) * chi(3*i*k*(n-i-k)/(i*k+k*(n-i-k)+i*(n-i-k))), where chi(n) = 1 - ceiling(n) + floor(n).

a(n) = A005044(n) - A337087(n).

Example

a(6) = 1; There is one integer-sided triangle with perimeter 6, [2,2,2]. The harmonic mean of its side lengths is 3*2*2*2/(2*2+2*2+2*2) = 2 (an integer).
a(10) = 1; There are 2 integer-sided triangles with perimeter 10, [2,4,4] and [3,3,4]. For the harmonic mean of the [2,4,4] triangle, we get 3*2*4*4/(2*4+2*4+4*4) = 96/32 = 3 (an integer), but the harmonic mean for the [3,3,4] triangle is 3*3*3*4/(3*3+3*4+3*4) = 108/33 (not an integer). Thus, a(10) = 1.

Mathematica

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

Crossrefs

Cf. A005044, A337087.

Sequence in context: A364048 A353657 A259896 * A113313 A342708 A074871

Adjacent sequences: A337083 A337084 A337085 * A337087 A337088 A337089

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Aug 14 2020

Status

approved