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%I #15 Jan 22 2016 04:35:15
%S 0,0,0,0,1,0,1,1,1,1,2,1,3,3,2,2,5,3,5,4,4,4,7,4,7,7,6,6,10,6,9,9,9,
%T 11,10,7,12,11,12,10,16,10,16,13,11,15,18,13,18,17,17,16,23,16,17,16,
%U 18,20,24,17,26,25,21,22,24
%N Number of triples (x,y,z) of integers such that 0 <= x <= y < n, 0 <= z < n, n^2 = x^2 + y^2 - z^2.
%C For any integer n>=7, there exist integers x, y, z such that 0 <= x < n, 0 <= y < n, 0 <= z < n, n^2 = x^2 + y^2 - z^2. Hence the only functions f of the positive integers into themselves such that f(m^2+n^2) = f(m)^2 + f(n)^2 are the identity and the null function.
%t r[n_] := Reduce[n^2 == x^2 + y^2 - z^2 && 0 <= x <= y < n && 0 <= z < n, {x, y, z}, Integers]; Reap[For[n = 1, n <= 100, n++, rn = r[n]; w = Which[rn === False, 0, Head[rn] === And, 1, Head[rn] === Or, Length[rn], True, -1; Print[n, " error"]]; Print[w]; Sow[w]]][[2, 1]] (* _Jean-François Alcover_, Jan 21 2016 *)
%o (Turbo-Pascal) VAR nMax, n,nbt,x,y,z: integer; BEGIN write('Maximum value of n: '); readln(nMax); for n := 1 to nMax do begin write(n,':'); nbt := 0; for y := 0 to n-1 do for x := 0 to y do for z := 0 to n-1 do begin if x*x+y*y-z*z=n*n then begin nbt := nbt+1; write('(',x,',',y,',',z,')'); end; end; writeln(' ',nbt,'.'); end; END.
%K nonn
%O 1,11
%A _Roger Cuculière_, Dec 10 2003
%E More terms from _Jean-François Alcover_, Jan 21 2016