This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A100450 #45 Oct 18 2019 10:23:37
%S A100450 1,7,19,51,99,195,291,483,675,963,1251,1731,2115,2787,3363,4131,4899,
%T A100450 6051,6915,8355,9507,11043,12483,14595,16131,18531,20547,23139,25443,
%U A100450 28803,31107,34947,38019,41859,45315,49923,53379,58851,63171,68547
%N A100450 Number of ordered triples (i,j,k) with |i| + |j| + |k| <= n and gcd(i,j,k) <= 1.
%C A100450 Note that gcd(0,m) = m for any m.
%C A100450 I would also like to get the sequences of the numbers of distinct sums i+j+k (also distinct products i*j*k) over all ordered triples (i,j,k) with |i| + |j| + |k| <= n; also over all ordered triples (i,j,k) with |i| + |j| + |k| <= n and gcd(i,j,k) <= 1.
%C A100450 Also the sequences of the numbers of distinct sums i+j+k (also distinct products i*j*k) over all ordered triples (i,j,k) with i >= 0, j >= 0, k >= 0 and i + j + k = n; also over all ordered triples (i,j,k) with i >= 0, j >= 0, k >= 0, i + j + k = n and gcd(i,j,k) <= 1.
%C A100450 Also the number of ordered triples (i,j,k) with i >= 0, j >= 0, k >= 0, i + j + k = n and gcd(i,j,k) <= 1.
%C A100450 From _Robert Price_, Mar 05 2013: (Start)
%C A100450 The sequences that address the previous comments are:
%C A100450 Distinct sums i+j+k with or without the GCD qualifier results in a(n)=2n+1 (A005408).
%C A100450 Distinct products i*j*k without the GCD qualifier is given by A213207.
%C A100450 Distinct products i*j*k with the GCD qualifier is given by A213208.
%C A100450 With the restriction i,j,k >= 0 ...
%C A100450 Distinct sums or products equal to n is trivial and always equals one (A000012).
%C A100450 Distinct sums <= n results in a(n)=n (A001477).
%C A100450 Distinct products <= n without the GCD qualifier is given by A213213.
%C A100450 Distinct products <= n with the GCD qualifier is given by A213212.
%C A100450 Ordered triples with sum = n without the GCD qualifier is A000217(n+1).
%C A100450 Ordered triples with sum = n with the GCD qualifier is A048240.
%C A100450 Ordered triples with sum <= n without the GCD qualifier is A000292.
%C A100450 Ordered triples with sum <= n with the GCD qualifier is A048241. (End)
%C A100450 This sequence (A100450) without the GCD qualifier results in A001845. - _Robert Price_, Jun 04 2013
%H A100450 Alois P. Heinz, <a href="/A100450/b100450.txt">Table of n, a(n) for n = 0..1000</a>
%F A100450 G.f.: (3 + Sum_{k>=1} (moebius(k)*((1+x^k)/(1-x^k))^3))/(1-x). - _Vladeta Jovovic_, Nov 22 2004. [Sketch of proof: Let b(n) = number of ordered triples (i, j, k) with |i| + |j| + |k| = n and gcd(i, j, k) <= 1. Then a(n) = A100450(n) = partial sums of b(n) and Sum_{d divides n} b(d) = 4*n^2+2 = A005899(n) with g.f. ((1+x)/(1-x))^3.]
%p A100450 f:=proc(n) local i,j,k,t1,t2,t3; t1:=0; for i from -n to n do for j from -n to n do t2:=gcd(i,j); for k from -n to n do if abs(i) + abs(j) + abs(k) <= n then t3:=gcd(t2,k); if t3 <= 1 then t1:=t1+1; fi; fi; od: od: od: t1; end;
%t A100450 f[n_] := Length[ Union[ Flatten[ Table[ If[ Abs[i] + Abs[j] + Abs[k] <= n && GCD[i, j, k] <= 1, {i, j, k}, {0, 0, 0}], {i, -n, n}, {j, -n, n}, {k, -n, n}], 2]]]; Table[ f[n], {n, 0, 40}] (* _Robert G. Wilson v_, Dec 14 2004 *)
%Y A100450 Cf. A000124, A000292, A018805, A027430, A048240, A048241, A100448, A100449, A213207, A213208, A213212, A213213.
%K A100450 nonn,easy
%O A100450 0,2
%A A100450 _N. J. A. Sloane_, Nov 21 2004