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, ...

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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.

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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, ...

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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]

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Best,

Ι.