A308109 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, 0, 0, 0, 0, 0, 0, 3, 0, 4, 4, 9, 5, 16, 11, 24, 19, 34, 28, 52, 39, 66, 59, 89, 74, 116, 99, 145, 128, 178, 159, 224, 194, 264, 243, 318, 285, 378, 342, 441, 405, 510, 471, 597, 543, 676, 634, 774, 716, 880, 818, 990, 928, 1108, 1042, 1249, 1164, 1380

Offset

1,9

Links

Table of n, a(n) for n=1..59.

Wikipedia, Integer Triangle

Formula

a(n) = Sum_{k=1..floor((n-1)/3)} Sum_{i=k+1..floor((n-k-1)/2)} sign(floor((i+k)/(n-i-k+1))) * i.

Conjectures from Colin Barker, May 13 2019: (Start)

G.f.: x^9*(3 + 3*x + 4*x^2 + 2*x^3 + x^4) / ((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 + 1, Floor[(n - k - 1)/2]}], {k, Floor[(n - 1)/3]}], {n, 100}]

Crossrefs

Cf. A308096.

Sequence in context: A201563 A016644 A240966 * A354783 A209132 A019789

Adjacent sequences: A308106 A308107 A308108 * A308110 A308111 A308112

Keyword

nonn

Author

Wesley Ivan Hurt, May 13 2019

Status

approved