A308236 Take apart the sides of each of the integer-sided 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, 6, 0, 16, 24, 62, 42, 154, 130, 304, 280, 530, 504, 990, 820, 1448, 1452, 2260, 2040, 3318, 3080, 4634, 4398, 6256, 6006, 8674, 7952, 11046, 10840, 14424, 13608, 18402, 17544, 22980, 22128, 28248, 27360, 35208, 33330, 42040, 41202, 50864, 48840, 60796

Offset

1,3

Links

Table of n, a(n) for n=1..45.

Wikipedia, Integer Triangle

Formula

a(n) = 2 * Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/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^3*(3 + 6*x + 14*x^2 + 25*x^3 + 50*x^4 + 69*x^5 + 92*x^6 + 81*x^7 + 73*x^8 + 53*x^9 + 41*x^10 + 22*x^11 + 11*x^12) / ((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)

Mathematica

Table[2*Sum[Sum[(i*k + i*(n - i - k) + k*(n - i - k))*Sign[Floor[(i + k)/(n - i - k + 1)]], {i, k, Floor[(n - k)/2]}], {k, Floor[n/3]}], {n, 100}]

Crossrefs

Cf. A308235.

Sequence in context: A270536 A278712 A057401 * A019134 A167298 A242838

Adjacent sequences: A308233 A308234 A308235 * A308237 A308238 A308239

Keyword

nonn

Author

Wesley Ivan Hurt, May 16 2019

Status

approved