A070086 Areas of integer triangles [A070080(n), A070081(n), A070082(n)], rounded values.

0, 1, 2, 1, 2, 3, 2, 3, 4, 4, 4, 2, 4, 4, 6, 5, 6, 7, 3, 5, 5, 7, 8, 6, 7, 8, 9, 3, 6, 6, 9, 7, 10, 11, 7, 9, 10, 11, 12, 4, 6, 8, 10, 8, 12, 12, 14, 8, 10, 12, 13, 12, 15, 16, 4, 7, 9, 12, 10, 14, 10, 15, 16, 17, 9, 12, 13, 15, 14, 17, 18, 19, 5, 8, 10

Offset

1,3

Comments

Triangles [A070080(A070142(n)), A070081(A070142(n)), A070082(A070142(n))] have integer areas = a(A070142(k)) = A070149(k).

Links

Jean-François Alcover, Table of n, a(n) for n = 1..972

Eric Weisstein's World of Mathematics, Heron's Formula.

Reinhard Zumkeller, Integer-sided triangles

Formula

a(n) = sqrt(s*(s-u)*(s-v)*(s-w)), where u=A070080(n), v=A070081(n), w=A070082(n) and s = A070083(n)/2 = (u+v+w)/2.

Example

[A070080(25), A070081(25), A070082(25)] = [3,5,6] and s = A070083(25)/2 = (3+5+6)/2 = 7: a(25) = sqrt(s*(s-3)*(s-5)*(s-6)) = sqrt(7*(7-3)*(7-5)*(7-6)) = sqrt(7*4*2*1) = sqrt(56) = 7.48331, rounded = 7.

Mathematica

m = 50; (* max perimeter *)
sides[per_] := Select[Reverse /@ IntegerPartitions[per, {3}, Range[ Ceiling[per/2]]], #[[1]] < per/2 && #[[2]] < per/2 && #[[3]] < per/2 &];
triangles = DeleteCases[Table[sides[per], {per, 3, m}], {}] // Flatten[#, 1]& // SortBy[Total[#] m^3 + #[[1]] m^2 + #[[2]] m + #[[1]]&];
area[{a_, b_, c_}] := With[{p = (a+b+c)/2}, Sqrt[p(p-a)(p-b)(p-c)] // Round];
area /@ triangles (* Jean-François Alcover, Oct 03 2021 *)

Crossrefs

Cf. A051516, A055595, A069596, A069594, A046131.

Sequence in context: A008611 A025798 A161064 * A211990 A162618 A036576

Adjacent sequences: A070083 A070084 A070085 * A070087 A070088 A070089

Keyword

nonn

Author

Reinhard Zumkeller, May 05 2002

Status

approved