A105659 Number of different characteristics, this is the squarefree part of (a+b+c)(a+b-c)(a-b+c)(-a+b+c), of integral triangles (a,b,c) with diameter n.

1, 2, 4, 5, 8, 11, 12, 16, 18, 22, 28, 28, 35, 38, 49, 50, 57, 65, 75, 74, 87, 83, 112, 111, 114, 120, 135, 146, 175, 168, 196, 185, 213, 222, 219, 234, 267, 270, 293, 306, 339, 333, 386, 348, 365, 420, 460, 431, 445, 436, 490, 480, 577, 511, 549, 559, 610, 635

Offset

1,2

Links

Table of n, a(n) for n=1..58.

Example

a(3)=4 because the integral triangles with diameter 3 are (3,2,2), (3,3,1), (3,3,2), (3,3,3) and the characteristics are 7, 35, 2, 3.

Mathematica

SquareFreePart[n_] := Times @@ Apply[Power, ({#1[[1]], Mod[#1[[2]], 2]} & ) /@ FactorInteger[n], {1}]; SquareFreePart[{a_, b_, c_}] := SquareFreePart[ (a+b+c)*(a+b-c)*(a-b+c)*(-a+b+c)]; ok[{a_, b_, c_}] := a-b < c < a+b && a-c < b < a+c && b-c < a < b+c; triangles[a_] := Reap[Do[ If[ok[{a, b, c}], Sow[{a, b, c}]], {b, 1, a}, {c, 1, b}]][[ 2, 1]]; a[n_] := Length[ Union[ SquareFreePart /@ triangles[n]]]; Table[a[n], {n, 1, 58}] (* Jean-François Alcover, Sep 11 2012 *)

Crossrefs

Sequence in context: A322125 A174803 A282434 * A298990 A360445 A191169

Adjacent sequences: A105656 A105657 A105658 * A105660 A105661 A105662

Keyword

nonn

Author

Sascha Kurz, May 04 2005

Status

approved