A337087 Number of integer-sided triangles with perimeter n where the harmonic mean of the side lengths is not an integer.

0, 0, 0, 0, 1, 0, 2, 1, 1, 1, 4, 2, 4, 4, 6, 5, 8, 5, 10, 7, 11, 10, 14, 11, 16, 13, 17, 15, 21, 17, 23, 21, 26, 24, 29, 24, 32, 30, 35, 32, 40, 35, 44, 40, 44, 43, 51, 46, 56, 51, 60, 54, 64, 59, 69, 63, 74, 70, 80, 73, 84, 78, 88, 84, 95, 90, 102, 96, 107, 100, 113, 105, 119, 113

Offset

1,7

Links

Table of n, a(n) for n=1..74.

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))) * (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) - A337086(n).

Example

a(6) = 0; 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 (which is an integer), so a(6) = 0.
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[(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, A337086.

Sequence in context: A305502 A308211 A140130 * A321312 A327502 A354090

Adjacent sequences: A337084 A337085 A337086 * A337088 A337089 A337090

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Aug 14 2020

Status

approved