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%I #29 Dec 01 2017 03:00:00
%S 0,3,3,1,0,6,3,0,0,3,6,3,0,6,6,0,0,9,3,3,0,6,3,0,0,6,12,3,0,12,6,0,0,
%T 6,9,6,0,6,9,0,0,15,6,3,0,6,6,0,0,6,12,6,0,12,9,0,0,6,6,9,0,12,12,0,0,
%U 18,12,3,0,12,6,0,0,9,18,6,0,12,6,0,0,9,9,9
%N Number of nonnegative solutions to (x,y,z) = 1 and x^2 + y^2 + z^2 = n.
%C a(n)=0 for n in A047536. - _Robert Israel_, Nov 30 2017
%H Robert Israel, <a href="/A295848/b295848.txt">Table of n, a(n) for n = 0..10000</a> (n=0..200 from Seiichi Manyama)
%e a(1) = 3;
%e (1,0,0) = 1 and 1^2 + 0^2 + 0^2 = 1.
%e (0,1,0) = 1 and 0^2 + 1^2 + 0^2 = 1.
%e (0,0,1) = 1 and 0^2 + 0^2 + 1^2 = 1.
%e a(2) = 3;
%e (1,1,0) = 1 and 1^2 + 1^2 + 0^2 = 2.
%e (1,0,1) = 1 and 1^2 + 0^2 + 1^2 = 2.
%e (0,1,1) = 1 and 0^2 + 1^2 + 1^2 = 2.
%e a(3) = 1;
%e (1,1,1) = 1 and 1^2 + 1^2 + 1^2 = 3.
%e a(5) = 6;
%e (2,1,0) = 1 and 2^2 + 1^2 + 0^2 = 5.
%e (2,0,1) = 1 and 2^2 + 0^2 + 1^2 = 5.
%e (1,2,0) = 1 and 1^2 + 2^2 + 0^2 = 5.
%e (1,0,2) = 1 and 1^2 + 0^2 + 2^2 = 5.
%e (0,2,1) = 1 and 0^2 + 2^2 + 1^2 = 5.
%e (0,1,2) = 1 and 0^2 + 1^2 + 2^2 = 5.
%p N:= 100:
%p V:= Array(0..N):
%p for x from 0 to floor(sqrt(N/3)) do
%p for y from x to floor(sqrt((N-x^2)/2)) do
%p for z from y to floor(sqrt(N-x^2-y^2)) do
%p if igcd(x,y,z) = 1 then
%p r:= x^2 + y^2 + z^2;
%p m:= nops({x,y,z});
%p if m=3 then V[r]:= V[r]+6
%p elif m=2 then V[r]:= V[r]+3
%p else V[r]:= V[r]+1
%p fi
%p fi
%p od od od:
%p convert(V,list); # _Robert Israel_, Nov 30 2017
%t f[n_] := Total[ Length@ Permutations@# & /@ Select[ PowersRepresentations[n, 3, 2], GCD[#[[1]], #[[2]], #[[3]]] == 1 &]]; Array[f, 90, 0] (* _Robert G. Wilson v_, Nov 30 2017 *)
%Y Cf. A002102, A047536, A048240, A295819, A295849.
%K nonn,easy,look
%O 0,2
%A _Seiichi Manyama_, Nov 29 2017
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