A336973 Sum of the smallest and largest side lengths of all distinct integer-sided triangles with perimeter n.

0, 0, 2, 0, 3, 4, 9, 5, 17, 13, 28, 23, 41, 36, 67, 51, 86, 81, 121, 102, 160, 141, 205, 184, 254, 233, 327, 286, 387, 365, 474, 429, 567, 522, 669, 621, 777, 729, 920, 843, 1044, 994, 1206, 1124, 1376, 1294, 1558, 1472, 1748, 1662, 1984, 1860, 2195, 2106, 2455, 2325, 2725, 2595

Offset

1,3

Links

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

Wikipedia, Integer Triangle

Index entries for linear recurrences with constant coefficients, signature (-1,0,2,4,2,-1,-5,-5,-1,2,4,2,0,-1,-1).

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 - i).

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

G.f.: x^3*(2 + 2*x + 3*x^2 + 3*x^3 + 5*x^4 + 4*x^5 + 4*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 and largest two side lengths is 1 + 1 = 2.
a(7) = 9; There are two integer-sided triangles with perimeter 7, [1,3,3] and [2,2,3]. The sum of the smallest and largest two side lengths of these triangles is 1 + 3 + 2 + 3 = 9.

Mathematica

Table[Sum[Sum[(n - i)*Sign[Floor[(i + k)/(n - i - k + 1)]], {i, k, Floor[(n - k)/2]}], {k, Floor[n/3]}], {n, 80}]
LinearRecurrence[{-1, 0, 2, 4, 2, -1, -5, -5, -1, 2, 4, 2, 0, -1, -1}, {0, 0, 2, 0, 3, 4, 9, 5, 17, 13, 28, 23, 41, 36, 67}, 60] (* Harvey P. Dale, Sep 22 2024 *)

Crossrefs

Cf. A005044.

Sequence in context: A110990 A254213 A321171 * A369017 A352846 A035347

Adjacent sequences: A336970 A336971 A336972 * A336974 A336975 A336976

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Aug 09 2020

Status

approved