A307686 Sum of the smallest side lengths of all integer-sided triangles with perimeter n.

0, 0, 1, 0, 1, 2, 3, 2, 6, 5, 9, 9, 13, 13, 22, 18, 27, 29, 38, 35, 51, 48, 64, 63, 79, 78, 103, 95, 120, 122, 147, 141, 177, 171, 207, 204, 240, 237, 286, 273, 322, 323, 372, 362, 426, 416, 480, 474, 538, 532, 613, 594, 675, 674, 755, 740, 840, 825, 925

Offset

1,6

Links

Table of n, a(n) for n=1..59.

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))) * k.

Conjectures from Colin Barker, May 12 2019: (Start)

G.f.: x^3*(1 + x + x^2 + x^3 + x^4 + x^5 + x^6) / ((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, Floor[(n - k)/2]}], {k, Floor[n/3]}], {n, 100}]

Crossrefs

Cf. A005044.

Sequence in context: A054126 A323507 A144176 * A077418 A005421 A348083

Adjacent sequences: A307683 A307684 A307685 * A307687 A307688 A307689

Keyword

nonn

Author

Wesley Ivan Hurt, May 11 2019

Status

approved