A308235 Take apart the sides of each of the integer-sided scalene triangles with perimeter n (at their vertices) and rearrange them orthogonally in 3-space so that their endpoints coincide at a single point. a(n) is the total surface area of all rectangular prisms enclosed in this way.

0, 0, 0, 0, 0, 0, 0, 0, 52, 0, 76, 94, 212, 126, 426, 328, 724, 624, 1130, 1020, 1938, 1540, 2648, 2568, 3910, 3432, 5482, 4970, 7364, 6850, 9616, 9072, 12954, 11696, 16086, 15576, 20544, 19152, 25698, 24240, 31530, 30072, 38148, 36630, 46870, 44022, 55240

Offset

1,9

Links

Table of n, a(n) for n=1..47.

Wikipedia, Integer Triangle

Formula

a(n) = 2 * 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*k + i*(n-i-k) + k*(n-i-k)).

Conjectures from Colin Barker, May 16 2019: (Start)

G.f.: 2*x^9*(26 + 52*x + 90*x^2 + 97*x^3 + 94*x^4 + 71*x^5 + 56*x^6 + 31*x^7 + 17*x^8 + 5*x^9 + x^10) / ((1 - x)^5*(1 + x)^4*(1 + x^2)^3*(1 + x + x^2)^3).

a(n) = -2*a(n-1) - 2*a(n-2) + a(n-3) + 7*a(n-4) + 10*a(n-5) + 7*a(n-6) - 5*a(n-7) - 17*a(n-8) - 19*a(n-9) - 9*a(n-10) + 9*a(n-11) + 19*a(n-12) + 17*a(n-13) + 5*a(n-14) - 7*a(n-15) - 10*a(n-16) - 7*a(n-17) - a(n-18) + 2*a(n-19) + 2*a(n-20) + a(n-21) for n>21.

(End)

Example

There is one integer-sided scalene triangle with perimeter 9: (2,3,4). The surface area of the enclosed rectangular prism is 2*(2*3 + 2*4 + 3*4) = 52. So a(9) = 52.

Mathematica

2*Sum[Sum[(i*k + i*(n - i - k) + k*(n - i - k))*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. A308233.

Sequence in context: A324683 A214373 A230299 * A022079 A230107 A275412

Adjacent sequences: A308232 A308233 A308234 * A308236 A308237 A308238

Keyword

nonn

Author

Wesley Ivan Hurt, May 16 2019

Status

approved