Values of Binary Partition Function Represented by a Sum of Three Squares

Abstract Let m be a positive integer and $$b_{m}(n)$$ b m ( n ) the number of partitions non-negative n with parts being powers 2, where each part can take colors. We show that if $$m=2^{k}-1$$ = 2 k - 1 , then natural density such cannot represented as sum three squares exists, equals 1/12 for $$k=1, 2$$ , 1/6 $$k\ge 3$$ ≥ 3 . In particular, $$m=1$$ equation $$b_{1}(n)=x^2+y^2+z^2$$ x + y z has solution in integers only is not form $$2^{2k+2}(8s+2t_{s}+3)+i$$ 8 s t i $$i=0, 1$$ 0 k s are integers, $$t_{n}$$ th term Prouhet–Thue–Morse sequence. A similar characterization obtained solutions $$b_{2^k-1}(n)=x^2+y^2+z^2$$