A336974 Sum of the largest two side lengths of all distinct integer-sided triangles with perimeter n.

0, 0, 2, 0, 4, 4, 11, 6, 21, 15, 35, 27, 52, 43, 83, 62, 109, 97, 152, 125, 201, 172, 258, 225, 321, 286, 410, 353, 489, 448, 597, 531, 714, 645, 843, 768, 981, 903, 1157, 1047, 1318, 1231, 1520, 1398, 1734, 1608, 1964, 1830, 2206, 2068, 2498, 2318, 2770, 2620, 3095, 2900, 3435, 3235

Offset

1,3

Links

Table of n, a(n) for n=1..58.

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))) * (n - k).

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

G.f.: x^3*(2 + 2*x + 4*x^2 + 4*x^3 + 7*x^4 + 5*x^5 + 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)

Example

a(3) = 2; There is one integer-sided triangle with perimeter 3, [1,1,1]. The sum of the largest two side lengths is 1 + 1 = 2.
a(7) = 11; There are two distinct integer-sided triangles with perimeter 7, [1,3,3] and [2,2,3]. The sum of the largest two side lengths of these triangles is 3 + 3 + 2 + 3 = 11.

Mathematica

Table[Sum[Sum[(n - 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: A021493 A195395 A296805 * A084247 A300307 A286606

Adjacent sequences: A336971 A336972 A336973 * A336975 A336976 A336977

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Aug 09 2020

Status

approved