A330915 Sum of the "middle" side lengths (b such that a <= b <= c) of all Heronian triangles with perimeter A051518(n).

4, 5, 5, 8, 12, 23, 45, 15, 29, 13, 48, 30, 77, 24, 69, 117, 25, 25, 46, 119, 20, 26, 110, 246, 26, 167, 172, 205, 169, 79, 468, 33, 229, 38, 222, 167, 429, 41, 429, 101, 270, 560, 416, 100, 153, 276, 390, 717, 50, 615, 61, 61, 60, 404, 634, 214, 130, 130, 1033, 975, 382

Offset

1,1

Links

Table of n, a(n) for n=1..61.

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))) * i, where chi(n) = 1 - ceiling(n) + floor(n) and c(n) = A051518(n). - Wesley Ivan Hurt, May 12 2020

Example

a(1) = 4; there is one Heronian triangle with perimeter A051518(1) = 12, which is [3,4,5] and its "middle" side length is 4.
a(6) = 23; there are two Heronian triangles with perimeter A051518(6) = 32, [4,13,15] and [10,10,12]. The sum is 13 + 10 = 23.

Crossrefs

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

Cf. A330912, A330916.

Sequence in context: A198334 A055593 A330923 * A268604 A246823 A088245

Adjacent sequences: A330912 A330913 A330914 * A330916 A330917 A330918

Keyword

nonn

Author

Wesley Ivan Hurt, May 02 2020

Status

approved