A330921 Sum of the areas of all Heronian triangles with perimeter A051518(n).

6, 12, 12, 24, 30, 72, 198, 60, 126, 66, 288, 180, 360, 84, 330, 648, 132, 204, 420, 876, 114, 156, 840, 1764, 264, 1350, 1632, 2016, 1830, 624, 3816, 330, 2604, 456, 2280, 2352, 4800, 780, 4422, 1224, 2940, 7068, 5430, 912, 2310, 3744, 5520, 9144, 984, 8736, 1020

Offset

1,1

Links

Table of n, a(n) for n=1..51.

Eric Weisstein's World of Mathematics, Heronian Triangle

Wikipedia, Heronian triangle

Wikipedia, Integer Triangle

Formula

a(n) = Sum_{k=1..floor(c(n)/3)} Sum_{i=k..floor((c(n)-k)/2)} sign(floor((i+k)/(c(n)-i-k+1))) * chi(sqrt((c(n)/2)*(c(n)/2-i)*(c(n)/2-k)*(c(n)/2-(c(n)-i-k)))) * sqrt((c(n)/2)*(c(n)/2-i)*(c(n)/2-k)*(c(n)/2-(c(n)-i-k)), where chi(n) = 1 - ceiling(n) + floor(n) and c(n) = A051518(n). - Wesley Ivan Hurt, May 12 2020

Example

a(1) = 6; there is one Heronian triangle with perimeter A051518(1) = 12, which is [3,4,5] and its area is 3*4/2 = 6.
a(6) = 72; there are two Heronian triangles with perimeter A051518(6) = 32, [4,13,15] and [10,10,12]. The sum of their areas 24 + 48 = 72.

Crossrefs

Cf. A051516, A051518, A070138, A096468, A298079, A298614, A305717.

Cf. A330912, A330915, A330916.

Sequence in context: A181798 A323332 A330922 * A183688 A298016 A055595

Adjacent sequences: A330918 A330919 A330920 * A330922 A330923 A330924

Keyword

nonn

Author

Wesley Ivan Hurt, May 02 2020

Status

approved