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A224442 Numbers that are the sum of three squares (square 0 allowed) in exactly two ways.

%I A224442 #25 Jan 15 2018 15:32:48
%S A224442 9,17,18,25,26,27,29,33,34,36,38,45,49,51,53,57,59,61,62,68,69,72,73,
%T A224442 75,77,82,83,85,94,97,100,102,104,105,106,107,108,109,116,118,123,130,
%U A224442 132,136,138,139,141,144,147,152,154,155,157,158,165,177,180,187
%N A224442 Numbers that are the sum of three squares (square 0 allowed) in exactly two ways.
%C A224442 These are the numbers for which A000164(a(n)) = 2.
%C A224442 a(n) is the n-th largest number which has a representation as sum of three integer squares (square 0 allowed), in exactly two ways, if neither the order of terms nor the signs of the numbers to be squared are taken into account. The multiplicity for a(n) with order and signs taken into account is A005875(a(n)).
%C A224442 This sequence is a proper subsequence of A000378.
%H A224442 Alois P. Heinz, <a href="/A224442/b224442.txt">Table of n, a(n) for n = 1..1000</a>
%F A224442 This sequence gives the increasingly ordered elements of the set {m integer | m = a^2 + b^2 + c^2, a, b and c integers with 0 <= a <= b <= c, and m has exactly two such representation}.
%F A224442 The sequence gives the increasingly ordered members of the set {m integer | A000164(m) = 2, m >= 0}.
%e A224442 a(1) = 9 = 0^2 + 0^2 + 3^2 = 1^2 + 2^2 + 2^2, and 9 is the smallest number m with A000164(m) = 2 for m >= 0. The multiplicity with order and signs taken into account is 2*3 + 8*3 = 30 = A005875(9).
%e A224442 The two representations [a,b,c] for a(n), n = 1, ..., 10, are
%e A224442 n=1,   9 = [0, 0, 3], [1, 2, 2],
%e A224442 n=2,  17 = [0, 1, 4], [2, 2, 3],
%e A224442 n=3,  18 = [0, 3, 3], [1, 1, 4],
%e A224442 n=4,  25 = [0, 0, 5], [0, 3, 4],
%e A224442 n=5,  26 = [0, 1, 5], [1, 3, 4],
%e A224442 n=6,  27 = [1, 1, 5], [3, 3, 3],
%e A224442 n=7,  29 = [0, 2, 5], [2, 3, 4],
%e A224442 n=8,  33 = [1, 4, 4], [2, 2, 5],
%e A224442 n=9,  34 = [0, 3, 5], [3, 3, 4],
%e A224442 n=10, 36 = [0, 0, 6], [2, 4, 4].
%p A224442 b:= proc(n, i, t) option remember; `if`(n=0, 1, `if`(t*i^2<n, 0,
%p A224442       `if`(b(n, i-1, t)>2, 3, min(3, b(n, i-1, t)+
%p A224442       `if`(i^2>n, 0, b(n-i^2, i, t-1))))))
%p A224442     end:
%p A224442 a:= proc(n) option remember; local k;
%p A224442       for k from 1 +`if`(n=1, 0, a(n-1))
%p A224442       while b(k, isqrt(k), 3)<>2 do od; k
%p A224442     end:
%p A224442 seq(a(n), n=1..100);  # _Alois P. Heinz_, Apr 09 2013
%t A224442 Select[ Range[0, 200], Length[ PowersRepresentations[#, 3, 2]] == 2 &] (* _Jean-François Alcover_, Apr 09 2013 *)
%Y A224442 Cf. A000164, A005875, A000378, A094942 (one way), A224443 (three ways).
%K A224442 nonn
%O A224442 1,1
%A A224442 _Wolfdieter Lang_, Apr 08 2013