cp's OEIS Frontend

A240227 All even positive integers which have at least one 'balanced' representation as a sum of three distinct nonzero squares.

%I A240227 #14 Feb 20 2015 06:46:15
%S A240227 14,26,38,42,56,62,74,78,86,98,104,114,122,126,134,146,152,158,168,
%T A240227 182,186,194,206,218,222,224,234,248,254,258,266,278,294,296,302,312,
%U A240227 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 A240227 All even positive integers which have at least one 'balanced' representation as a sum of three distinct nonzero squares.
%C A240227 For the numbers with representations as a sum of three distinct nonzero squares see A004432. For their multiplicity see A025442.
%C A240227 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 A240227 The multiplicities are given in A240228.
%C A240227 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 A240227 Wolfdieter Lang, <a href="/A240227/a240227.pdf">The first twenty representations.</a>
%F A240227 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 A240227 a(n) = 2*A024606(n). - _Robert Israel_, May 21 2014
%e A240227 n  a(n) (u, v=u+w, w)  [x, y,z]  A236300 member
%e A240227 1:  14   (1, 3, 2)    [4, 3, 1]     8*7 =   56
%e A240227 2:  26   (1, 4, 3)    [5, 4, 1]   10*13 =  130
%e A240227 3:  38   (2, 5, 3)    [7, 5, 2]   14*19 =  266
%e A240227 4:  42   (1, 5, 4)    [6, 5, 1]   12*21 =  252
%e A240227 5:  56   (2, 6, 4)    [8, 6, 2]   16*28 =  448
%e A240227 6:  62   (1, 6, 5)    [7, 6, 1]   14*31 =  434
%e A240227 7:  74   (3, 7, 4)   [10, 7, 3]   20*37 =  740
%e A240227 8:  78   (2, 7, 5)    [9, 7, 2]   18*39 =  702
%e A240227 9:  86   (1, 7, 6)    [8, 7, 1]   16*43 =  688
%e A240227 10: 98   (3, 8, 5)   [11, 8, 3]   22*49 = 1078 ...
%e A240227 For n=11 .. 20 see the link.
%Y A240227 Cf. A004432, A025442, A236300, A240228 (multiplicities).
%K A240227 nonn,easy
%O A240227 1,1
%A A240227 _Wolfdieter Lang_, May 02 2014