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

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, 41, 47, 48, 49, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 66, 76, 77, 78, 79, 84, 85, 87, 89, 90, 91, 92, 94, 96, 97, 98, 100, 102, 103, 104, 107, 109, 123, 124, 125, 126

Offset: 1

Author

Patrick Okolo Edeogu, Jul 10 2015

Keywords

• nonn

Comments

This sequence shows that the "proper digital expansion" mentioned in the introduction to the paper by Drmota and Gajdosik (see Links) is not unique.

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

Examples

			7 = 5 + 2 is unique with respect to its minimal digit sum 1 + 1 = 2.
But 10 = 8 + 2 = 5 + 5 is not unique with respect to its minimal digit sum 1 + 1 = 2.
		

Links

• Patrick Okolo Edeogu, Table of n, a(n) for n = 1..141
• M. Drmota and M. Gajdosik, The parity of the sum of digits function of generalized Zeckendorf expansions, The Fibonacci Quarterly, 36:1 (1988), pp. 3-19.

Crossrefs

Cf. A000045.

Programs

Maple

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);