A308096 Take all the integer-sided triangles with perimeter n and sides a, b, and c such that a <= b <= c. a(n) is the sum of all the b's.

0, 0, 1, 0, 2, 2, 5, 3, 10, 7, 16, 13, 24, 20, 38, 29, 50, 45, 69, 58, 92, 79, 117, 104, 146, 131, 186, 162, 222, 205, 270, 243, 324, 294, 381, 351, 444, 411, 523, 477, 596, 560, 686, 636, 784, 730, 886, 832, 996, 938, 1127, 1052, 1250, 1188, 1395, 1315

Offset

1,5

Links

Table of n, a(n) for n=1..56.

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.

Conjectures from Colin Barker, May 17 2019: (Start)

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

Crossrefs

Cf. A307686 (sum of smallest sides), A307966 (sum of largest sides).

Sequence in context: A331520 A160793 A327754 * A006800 A241820 A190170

Adjacent sequences: A308093 A308094 A308095 * A308097 A308098 A308099

Keyword

nonn

Author

Wesley Ivan Hurt, May 12 2019

Status

approved