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A256133 Numbers that have unique expansion with minimal digit sum in terms of Fibonacci numbers F_k (k > 1).

%I A256133 #47 Jan 05 2025 19:51:40
%S A256133 1,2,3,5,7,8,9,11,13,14,15,18,20,21,22,23,24,29,30,32,34,35,36,37,39,
%T A256133 41,47,48,49,52,54,55,56,57,58,60,62,63,64,66,76,77,78,79,84,85,87,89,
%U A256133 90,91,92,94,96,97,98,100,102,103,104,107,109,123,124,125,126
%N A256133 Numbers that have unique expansion with minimal digit sum in terms of Fibonacci numbers F_k (k > 1).
%C A256133 This sequence shows that the "proper digital expansion" mentioned in the introduction to the paper by Drmota and Gajdosik (see Links) is not unique.
%C A256133 This sequence consists of all positive integers that have Zeckendorf expansions not containing any ...1001... and not ending in ...101. Example: 20 is in and it has 20 = 13 + 5 + 2 with Zeckendorf expansion 101010, while 19 is not in and has 19 = 13 + 5 + 1 with Zeckendorf expansion 101001. - _Thomas Bier_, Oct 09 2015
%H A256133 Patrick Okolo Edeogu, <a href="/A256133/b256133.txt">Table of n, a(n) for n = 1..141</a>
%H A256133 M. Drmota and M. Gajdosik, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/36-1/drmota.pdf">The parity of the sum of digits function of generalized Zeckendorf expansions</a>, The Fibonacci Quarterly, 36:1 (1988), pp. 3-19.
%e A256133 7 = 5 + 2 is unique with respect to its minimal digit sum 1 + 1 = 2.
%e A256133 But 10 = 8 + 2 = 5 + 5 is not unique with respect to its minimal digit sum 1 + 1 = 2.
%p A256133 x0:=0: x1:=1: ML:=[]: L:=[]: mes:=0:  for r from 2 to 14 do: z:=x1+x0: x0:=x1: x1:=z:  rj:=12: L:=[op(L),z]: ML:=[z,op(ML)]: od: XL:=[]: for m from 1 to 400 do:  NL:=[]: n:=m: for j from 12 to 1 by -1 do: if L[j+1]>n and L[j]-1 < n then NL:=[op(NL),j]: n:=n-L[j]: fi: od: eps:=0: nx:=nops(NL):  for j from 1 to nx-1 do: if NL[j]-NL[j+1]=3 then eps:=1: fi: if NL[nx-1]-NL[nx]=2 and NL[nx]=1 then eps:=1: fi:od: if eps=0 then XL:=[op(XL),m]: fi: od: print(XL);
%Y A256133 Cf. A000045.
%K A256133 nonn
%O A256133 1,2
%A A256133 _Patrick Okolo Edeogu_, Jul 10 2015