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A051516 Number of triangles with perimeter n having integer sides and area. 15
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 0, 4, 0, 0, 0, 1, 0, 2, 0, 1, 0, 0, 0, 3, 0, 2, 0, 0, 0, 4, 0, 1, 0, 0, 0, 3, 0, 0, 0, 5, 0, 1, 0, 1, 0, 2, 0, 5, 0, 0, 0, 1, 0, 1, 0, 4, 0, 0, 0, 8, 0, 0, 0, 1, 0, 5, 0, 0, 0, 0, 0, 5, 0, 6, 0, 5, 0, 0, 0, 2, 0, 0, 0, 12, 0, 1, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,32

COMMENTS

No such triangles with odd perimeter exist.

LINKS

Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 1000 terms from Seiichi Manyama)

Eric Weisstein's World of Mathematics, Heronian Triangle.

FORMULA

a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} chi(m) * sign(floor((i+k)/(n-i-k+1))), where m = sqrt((n/2)*(n/2-i)*(n/2-k)*(i+k-n/2)) and chi is the integer characteristic. - Wesley Ivan Hurt, May 11 2019

MATHEMATICA

Table[Sum[Sum[(1 - Ceiling[Sqrt[(n/2) (n/2 - i) (n/2 - k) (i + k - n/2)]] + Floor[Sqrt[(n/2) (n/2 - i) (n/2 - k) (i + k - n/2)]])*Sign[Floor[(i + k)/(n - i - k + 1)]], {i, k, Floor[(n - k)/2]}], {k, Floor[n/3]}], {n, 100}] (* Wesley Ivan Hurt, May 11 2019 *)

CROSSREFS

Cf. A024153, A070139.

Sequence in context: A287871 A135416 A134309 * A236799 A208274 A208604

Adjacent sequences:  A051513 A051514 A051515 * A051517 A051518 A051519

KEYWORD

nonn

AUTHOR

David W. Wilson

STATUS

approved

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Last modified April 9 07:30 EDT 2020. Contains 333344 sequences. (Running on oeis4.)