Representation of a 2-power as sum of k 2-powers: A recursive formula

For every integer k, a k-representation of 2k−1 is a string n=(n1,…,nk) of nonnegative integers such that , and W(1,k) is their number. We present an efficient recursive formula for W(1,k); this formula allows also to prove the congruence for k⩾3.