Infinite Motel

[Eric Angelini]

1,2,13,3,6,26,4,11,408,9,5,24,7,51,22,102,20,49,18,8,205,
10,16,12,47,14,100,45,[?]

The Infinite Motel doors seq A109817 raises a few
questions:
- what is the next term?
- will some doors be unlabelled for ever?
- could someone compute the reverse sequence [a(n) would
  be the label on the nth door]; the seq starts like that:

  1,2,4,7,11,5,13,20,10,22...
  which reads:
  "Label "1" is in position 1 in A109817
  "Label "2" is in position 2 in A109817
  "Label "3" is in position 4 in A109817
  "Label "4" is in position 7 in A109817
  "Label "5" is in position 11 in A109817
  "Label "6" is in position 5 in A109817
  "Label "7" is in position 13 in A109817
  "Label "8" is in position 20 in A109817
  "Label "9" is in position 10 in A109817
  "Label "10" is in position 22 in A109817...

- If you need one step to go from a door to the next one,
what is the sequence formed by the cumulated steps of the
labelling person in A109817? (start 1 step before door #1)

  Seq would start like that:
  1,2,4,7,11,17,25,32,42,54,68,84... which reads:

  1 step  to go from label "0" to label "1": S=1
  1 step  to go from label "1" to label "2": S=2
  2 steps to go from label "2" to label "3"; S=4
  3 steps to go from label "3" to label "4"; S=7
  4 steps to go from label "4" to label "5"; S=11
  6 steps to go from label "5" to label "6"; S=17
  8 steps to go from label "6" to label "7"; S=25
  7 steps to go from label "7" to label "8"; S=32
  10 steps to go from labl "8" to label "9"; S=42
...

Thanks,
best,
É.









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