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
Keywords
Examples
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.
Programs
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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 *)