A308102 Sum of the perimeters of all integer-sided scalene triangles with perimeter n.

0, 0, 0, 0, 0, 0, 0, 0, 9, 0, 11, 12, 26, 14, 45, 32, 68, 54, 95, 80, 147, 110, 184, 168, 250, 208, 324, 280, 406, 360, 496, 448, 627, 544, 735, 684, 888, 798, 1053, 960, 1230, 1134, 1419, 1320, 1665, 1518, 1880, 1776, 2156, 2000, 2448, 2288, 2756, 2592

Offset

1,9

Links

Table of n, a(n) for n=1..54.

Wikipedia, Integer Triangle

Formula

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

Conjectures from Colin Barker, May 13 2019: (Start)

G.f.: x^9*(3 + 2*x + x^2)*(3 + x + 2*x^2) / ((1 - x)^4*(1 + x)^3*(1 + x^2)^2*(1 + x + x^2)^2).

a(n) = -a(n-1) + 2*a(n-3) + 4*a(n-4) + 2*a(n-5) - a(n-6) - 5*a(n-7) - 5*a(n-8) - a(n-9) + 2*a(n-10) + 4*a(n-11) + 2*a(n-12) - a(n-14) - a(n-15) for n>15.

(End)

Mathematica

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

Prog

(PARI) a(n) = n * sum(k=1, (n-1)\3, sum(i=k+1, (n-k-1)\2, sign((i+k)\(n-i-k+1)))); \\ Michel Marcus, May 13 2019

Crossrefs

Cf. A005044.

Sequence in context: A341486 A349825 A320379 * A338016 A062047 A117465

Adjacent sequences: A308099 A308100 A308101 * A308103 A308104 A308105

Keyword

nonn

Author

Wesley Ivan Hurt, May 12 2019

Status

approved