A070148 Numbers k such that [A070080(k), A070081(k), A070082(k)] is an integer Heronian triangle having triangular area.

17, 368, 659, 972, 1156, 1599, 1971, 2555, 2574, 3746, 3818, 4298, 4330, 5374, 14325, 14414, 15004, 15943, 16451, 19475, 19615, 24013, 24051, 33950, 63593, 71630, 75052, 79286, 79670, 79921, 84183, 90187, 93290

Offset

1,1

Links

Jean-François Alcover, Table of n, a(n) for n = 1..89

Eric Weisstein's World of Mathematics, Heronian Triangle.

Eric Weisstein's World of Mathematics, Right Triangle.

Reinhard Zumkeller, Integer-sided triangles

Example

17 is a term: [A070080(17), A070081(17), A070082(17)] = [3,4,5]: A070086(52)=6.

Mathematica

m = 500 (* 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)]];
Position[triangles, {a_, b_, c_} /; IntegerQ[area[{a, b, c}]] && IntegerQ[Sqrt[1 + 8 area[{a, b, c}]]]] // Flatten (* Jean-François Alcover, Oct 04 2021 *)

Crossrefs

Cf. A070142, A000217.

Sequence in context: A293691 A365678 A002197 * A097499 A132541 A159244

Adjacent sequences: A070145 A070146 A070147 * A070149 A070150 A070151

Keyword

nonn

Author

Reinhard Zumkeller, May 05 2002

Status

approved