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50 Chapter 2. On a conjecture about addition modulo 2 k − 1Figure 2.5: Graph of f d (β i ) for β i = 10, i ≠ 1we prove this in Proposition 2.5.19. There is a recursion relation between coefficients with differentd’s:a d,n+1(i + 1,...,i n,0) ad,n (i = 1,...,i n) 3ad−1,n (i ;1,...,i n)this is Corollary 2.5.18. There is a relation between coefficients with a given d:this is Proposition 2.5.3.a d,n(i,j,...) = i + 1 a d,nj(i+1,j−1,...) ;2.5.2 Splitting the sum into atomic partsWe consider a general d ≥ 1. From Proposition 2.4.10, we havef d (β 1 , . . . , β d ) =B−1∑E=0∑∑d ei=E0≤e i∏P (e i ) ,where P (e i ) has three different expressions according to the value of e i :⎧⎪⎨ 2 −βi if e i = 0 ,2P (e i ) =−β i3⎪⎩(2ei − 2 −ei ) if 0 < e i < β i ,2 β i −2 −β i32 −ei if β i ≤ e i .Let us denote for a vector X ∈ {0, 1, 2} d :• the i-th coordinate by X i with 1 ≤ i ≤ d;• j k = # {i | X i = k} for 0 ≤ k ≤ 2;• B 0,1 = ∑ {i|X i≠2} β i;d