Complete Sequences, was Re: A054770

karttu at karttu at
Wed Jun 7 22:58:53 CEST 2000

> But A54770 clearly stated that it used the Lucas numbers in A203, not A24.

Dear Jud,

I admit I have recently been particularly unclear. The assumption in
my mind was simply, that, although the Lucas numbers
starting from 1 (A000204) are not complete, at least when
the "missing" 2 is added to them (A000032), they will be
complete then. It was only in this sense that I told Antreas
that he had "forgotten" the 2.
  Heck, the missing element might even be another 1 instead of 2,
although then the summands are no more distinct. In any case, for now
these are just hypotheses waiting a proof or more light from
that article of Zeckendorf himself.
  See below for the definition of "Complete Sequence",
quoted from:

 Complete Sequence 
      A sequence of numbers  V = {v_n} is complete if every positive
      integer n is the sum of some subsequence of
	  V, i.e., there exist a_i = 0 or 1 such that 

        n = Sum_{i=1..inf} a_i*v_i

      (Honsberger 1985, pp. 123-126). The Fibonacci numbers are complete.
      In fact, dropping one number still leaves a complete sequence,
      although dropping two numbers does not
      (Honsberger 1985, pp. 123 and 126).  

Parhain terveisin,
  Best regards,

Antti Karttunen
E-mail: karttu at

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