1 chain  with 3 bits set
  4 chains with 4 bits set

Notation:
   @(.)  means 2^(.).
   p,q,r are free nonnegative integer variables.
   A,B,C,D  are also free nonnegative integer variables.
   *  in the index-list as a right sub_index indicates a doubling step.
   :  is a delimiter in the sub-index to refine this sub-index by a further 'value'.
         Such a 'value' is in general a polynomial of degree 0 or 1 in one variable
         [i.e. an affine form with integer coefficients and dimension less or equal 1].
         This 'value' indicates either in a doubling step how many doubling steps are here,
            or when using a doubling step in another index how many doubling steps are used there.
         If a doubling step is used with exactly as many doubling steps it was generated then this 
            delimiter and 'value' is optional [E.g. It may be left out in the left sub-index for every star-chain].

In all two-power-sum representations the two-power-exponents are always strictly monotonic decreasing and >= 0.


Shortest Addition Chains with 2 Small Steps

  0 3* @(p)+@(q)+@(r)
    A=p,  B=q,  C=r
    0                 1
    1( 0, *:A)        @(A)
    2( 1, 1:B)        @(A)+@(B)
    3( 2, 1:C)        @(A)+@(B)+@(C)

  1 4* @(p)+@(p-5)+@(p-6)+@(p-7)
    A=p-6
    0                 1
    1( 0, *:A)        @(A)
    2( 1, 1:A-1)      @(A)+@(A-1)
    3( 2, 1  )        @(A+1)+@(A-1)
    4( 3, *:3)        @(A+4)+@(A+2)
    5( 4, 3  )        @(A+4)+@(A+2)+@(A+1)+@(A-1)
    6( 5, *:1)        @(A+5)+@(A+3)+@(A+2)+@(A)
    7( 6, 5  )        @(A+6)+@(A+1)+@(A)+@(A-1)

  2 4* @(p)+@(p-3)+@(q)+@(q-1)
    A=p-3,  C=q-1
    0                 1
    1( 0, *:A)        @(A)
    2( 1, 1:A-1)      @(A)+@(A-1)
    3( 2, 1:C)        @(A)+@(A-1)+@(C)
    4( 3, 2  )        @(A+1)+@(A)+@(C)
    5( 4, *:1)        @(A+2)+@(A+1)+@(C+1)
    6( 5, 4  )        @(A+3)+@(A)+@(C+1)+@(C)

  3 4* @(p)+@(q)+@(r)+@(-p+q+r)
    A=q-1,  B=-p+q+r,  D=p-q-1
    0                 1
    1( 0, *:A)        @(A)
    2( 1, 1:B)        @(A)+@(B)
    3( 2, 1  )        @(A+1)+@(B)
    4( 3, *:D+1)      @(A+D+2)+@(B+D+1)
    5( 4, 3  )        @(A+D+2)+@(A+1)+@(B+D+1)+@(B)

  4 4* @(p)+@(q)+@(r)+@(-p+q+r+1)
    A=r,  B=-p+q+r+1,  D=p-r-2
    0                 1
    1( 0, *:A)        @(A)
    2( 1, 1:B)        @(A)+@(B)
    3( 2, 1  )        @(A+1)+@(B)
    4( 3, *:D+1)      @(A+D+2)+@(B+D+1)
    5( 4, 2  )        @(A+D+2)+@(B+D+1)+@(A)+@(B)