A308107 Sum of the smallest side lengths of all integer-sided scalene triangles with perimeter n.

0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 2, 3, 5, 3, 9, 7, 13, 12, 18, 17, 29, 23, 35, 36, 48, 43, 63, 58, 78, 75, 95, 92, 122, 111, 141, 141, 171, 162, 204, 195, 237, 231, 273, 267, 323, 306, 362, 360, 416, 402, 474, 460, 532, 522, 594, 584, 674, 650, 740, 735, 825

Offset

1,9

Links

Table of n, a(n) for n=1..61.

Wikipedia, Integer Triangle

Formula

a(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))) * k.

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

G.f.: x^9*(2 + 2*x + 2*x^2 + x^3) / ((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[Sum[Sum[k*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) = sum(k=1, (n-1)\3, sum(i=k+1, (n-k-1)\2, k*sign((i+k)\(n-i-k+1)))); \\ Michel Marcus, May 13 2019

Crossrefs

Cf. A307686.

Sequence in context: A028376 A059235 A261216 * A323167 A361651 A222753

Adjacent sequences: A308104 A308105 A308106 * A308108 A308109 A308110

Keyword

nonn

Author

Wesley Ivan Hurt, May 13 2019

Status

approved