OFFSET
1,3
LINKS
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
KEYWORD
nonn,easy
AUTHOR
Wesley Ivan Hurt, Aug 09 2020
STATUS
approved