Hello SeqFans,
Two Aronson-like finite sequences, one infinite and one impossible:
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1 2 3 4
5 6 7
8 9 10
11 12 13
14 15 16
Words
having an odd
number of letters
in this sentence are
in position one,
four, seven,
17 18
19 20 21 22 23 24
25
eleven, fourteen, sixteen, nineteen, twenty-one, twenty-two, twenty-three and twenty-four.
S(1) =
1,4,7,11,14,16,19,21,22,23,24.
(the seq can be shorten to 1,4,7,11,14,16,19 or 1,4,7,11,14,16,19,21 or 1,4,7,11,14,16,19,21,22
or 1,4,7,11,14,16,19,21,22,23 if the word “and” is omitted)
The seq could be extended too, of course, to
an infinite amount of terms:
1 2 3 4
5 6 7
8 9 10
11 12 13
14 15 16
Words
having an odd
number of letters
in this sentence are
in position one,
four, seven,
17 18
19 20 21 22 23 24
eleven, fourteen, sixteen, nineteen, twenty-one, twenty-two, twenty-three, twenty-five,
25
26 27 28
twenty-six, twenty-seven, twenty-eight, thirty-one, ...
S(1b) =
1,4,7,11,14,16,19,21,22,23,25,26,27,28,31, ...
----------------------
1 2
3 4 5
6 7 8
9 10 11 12
13 14 15
16
Words having
an even number of letters in this sentence are in position two, three, four,
17 18
19 20 21
22 23 24 25
26 27
five, six, eight, nine,
ten, twelve, thirteen, sixteen,
seventeen, twenty,
twenty-two,
28 29
twenty-three
and twenty-six.
S(2) =
2,3,4,5,6,8,9,10,12,13,16,17,20,22,23,26.
----------------------
1 2
3 4
The first, second, fourth,
(...) words of this sentence have an odd
number of letters.
NO ! (false)
1 2
3 4
The first, second, fifth,
(...) words of this sentence have an odd
number of letters.
NO ! (leads to a non-monotonic
seq.)
1 2
3 4
The first, second, sixth, (...)
words of this sentence have an odd number
of letters.
NO ! (leads also to a
non-monotonic seq.)
1 2
3 4
The first, second, seventh,
(...) words of this sentence have an odd
number of letters.
NO ! (leads again to a non-monotonic seq.)
1 2
3 4 5
6 7 8 9 10
The first, second, eighth,
eleventh, thirteenth, fourteenth,
fifteenth,
eighteenth, nineteenth,
11 12 13
14 15 16 17
twentieth, twenty-second, twenty-third, twenty-fifth, twenty-sixth, twenty-eighth, thirty-second,
18
thirty-third (...) words
of this sentence have an odd
number of letters.
S(3) = 1,2,8,11,13,14,15,18,19,20,22,23,25,26,28,32,33, ...
----------------------
1 2
3 4 5
6 7 8
9 10 11 12
The second, fifth, sixth, eighth, eleventh,
twelfth, thirteenth,
fifteenth, sixteenth, eighteenth,
nineteenth,
13 14 15
twenty-second, twenty-third, twenty-fourth, (...)
words of this sentence have an even number of letters.
s
t u v
w x y
z
S(4) =
2,5,6,8,11,12,13,15,16,18,19,22,23,24, ...
[but this sequence is forever impossible to
build: one will never find eight consecutive even
ordinals
to describe s, t, u, v, w, x, y and z]
----------------------
Best,
Ι.