A336972 Sum of the smallest two side lengths of all distinct integer-sided triangles with perimeter n.

0, 0, 2, 0, 3, 4, 8, 5, 16, 12, 25, 22, 37, 33, 60, 47, 77, 74, 107, 93, 143, 127, 181, 167, 225, 209, 289, 257, 342, 327, 417, 384, 501, 465, 588, 555, 684, 648, 809, 750, 918, 883, 1058, 998, 1210, 1146, 1366, 1306, 1534, 1470, 1740, 1646, 1925, 1862, 2150, 2055, 2390, 2290, 2635

Offset

1,3

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))) * (i + k).

Conjectures from Colin Barker, Aug 10 2020: (Start)

G.f.: x^3*(2 + 2*x + 3*x^2 + 3*x^3 + 4*x^4 + 3*x^5 + 3*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)

Example

a(3) = 2; There is one integer-sided triangle with perimeter 3, [1,1,1]. The sum of the smallest two side lengths is 1 + 1 = 2.
a(7) = 8; There are two distinct integer-sided triangles with perimeter 7, [1,3,3] and [2,2,3]. The sum of the smallest two side lengths of these triangles is 1 + 3 + 2 + 2 = 8.

Mathematica

Table[Sum[Sum[(i + k)*Sign[Floor[(i + k)/(n - i - k + 1)]], {i, k, Floor[(n - k)/2]}], {k, Floor[n/3]}], {n, 80}]

Crossrefs

Cf. A005044.

Sequence in context: A344059 A258871 A283528 * A110990 A254213 A321171

Adjacent sequences: A336969 A336970 A336971 * A336973 A336974 A336975

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Aug 09 2020

Status

approved