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%I #14 Feb 20 2015 06:46:15
%S 14,26,38,42,56,62,74,78,86,98,104,114,122,126,134,146,152,158,168,
%T 182,186,194,206,218,222,224,234,248,254,258,266,278,294,296,302,312,
%U 314,326,338,342,344,350,362,366,378,386,392,398,402,416,422,434,438,446,456,458,474,482,488,494,504,518,536,542,546,554,558,566,582,584
%N All even positive integers which have at least one 'balanced' representation as a sum of three distinct nonzero squares.
%C For the numbers with representations as a sum of three distinct nonzero squares see A004432. For their multiplicity see A025442.
%C Here only even numbers are considered, and a representation 2*m = a^2 + b^2 + c^2, a > b > c > 0 denoted by the triple (a,b,c), is called 'balanced' if a = b + c. E.g., 62 is represented by (6, 5, 1) and (7, 3, 2) but only (6, 5, 1) is balanced because 6 = 5 + 1.
%C The multiplicities are given in A240228.
%C These numbers a(n) play a role in the problem proposed in A236300: Find all numbers which are of the form (x + y + z)*(u^2 + v^2 + w^2)/2, x >= y >= z >= 0, where u = x-y, v = x-z, w = y-z, with u >= 0, v >=0, w >= 0, u - v + w = 0 and even u^2 + v^2 + w^2 >= 4. The special case (called in a comment on A236300 case (iib)) with distinct u, v, and w, each >=1, needs the numbers a(n) of the present sequence. If the triple is taken as (u, u+w, w) with u < w then the [x, y, z] values are [2*u+w, u+w, u] and the number from A236300 is (2*u+w)*(u^2 + w^2 + u*w) =(2*u+w)*a(n). If this number from A236300 has multiplicity A240228(n) >=2 then there are A240228(n) different values for [x, y, z] and corresponding different A236300 numbers.
%H Wolfdieter Lang, <a href="/A240227/a240227.pdf">The first twenty representations.</a>
%F The increasingly ordered elements of the set {2*k, k positive integer : 2*k = u^2 + (u+w)^2 + w^2 with 1 <= u < w }.
%F a(n) = 2*A024606(n). - _Robert Israel_, May 21 2014
%e n a(n) (u, v=u+w, w) [x, y,z] A236300 member
%e 1: 14 (1, 3, 2) [4, 3, 1] 8*7 = 56
%e 2: 26 (1, 4, 3) [5, 4, 1] 10*13 = 130
%e 3: 38 (2, 5, 3) [7, 5, 2] 14*19 = 266
%e 4: 42 (1, 5, 4) [6, 5, 1] 12*21 = 252
%e 5: 56 (2, 6, 4) [8, 6, 2] 16*28 = 448
%e 6: 62 (1, 6, 5) [7, 6, 1] 14*31 = 434
%e 7: 74 (3, 7, 4) [10, 7, 3] 20*37 = 740
%e 8: 78 (2, 7, 5) [9, 7, 2] 18*39 = 702
%e 9: 86 (1, 7, 6) [8, 7, 1] 16*43 = 688
%e 10: 98 (3, 8, 5) [11, 8, 3] 22*49 = 1078 ...
%e For n=11 .. 20 see the link.
%Y Cf. A004432, A025442, A236300, A240228 (multiplicities).
%K nonn,easy
%O 1,1
%A _Wolfdieter Lang_, May 02 2014
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