[seqfan] Two self-describing sequences

[Éric Angelini]

Hello Seqfans,
Here is a self-describing sequence – when translated into English:

S = 5,14,84,10,1,20,21,22,17,4,4,27,11,2,98,99,9,34,1,6,7,8,9,4,12,6,12,4,9,36,4,12,9,18,9,36,4,12,9,18,6,12,4,9,36,4,12,9,18,9,30,6,...

FIVE FOURTEEN EIGHTYFOUR TEN ONE TWENTY TWENTYONE TWENTYTWO SEVENTEEN FOUR FOUR TWENTYSEVEN ELEVEN TWO NINETYEIGHT NINETYNINE NINE THIRTYFOUR ONE SIX SEVEN EIGHT NINE FOUR TWELVE SIX TWELVE FOUR NINE THIRTYSIX FOUR TWELVE NINE EIGHTEEN NINE THIRTYSIX FOUR TWELVE NINE EIGHTEEN SIX TWELVE FOUR NINE THIRTYSIX FOUR TWELVE NINE EIGHTEEN NINE THIRTY SIX...

Duplicate hereunder the 5th letter of the sequence:

F

Go on with the 14th:

FI

And the 84th:

FIV

The 10th:

FIVE

The 1st:

FIVE F

The 20th: 

FIVE FO

Etc.

I’ve tried hard but I’m quite sure this is not the lexicographically earliest sequence of this kind [a(3) might be lowered, I guess].

The true first sequence of this kind is of no interest:

ONE TWO THREE FOUR FIVE SIX SEVEN... ad lib.

We thus decided that a term of S cannot ask the duplication of one of its own letters – the letter to be duplicated must be found elsewhere, before or after [this is why the F of FIVE is given by the F of FOURTEEN (in position 5), the I of FIVE by the I of EIGHTYFOUR (in position 14), the V of FIVE by the V of ELEVEN (in position 84) and the E of FIVE by the E of FOURTEEN (in position 10, the next term)].

The same self-description idea works if one is asked to duplicate digits whose successive positions in the sequence are determined by the integers themselves (the same “don’t use me to describe myself” rule applies):

T = 2,20,6,10,8,11,60,5,13,4,3,19,52,23,30,14,9,...

(If I’m not mistaken, this should be the lexicographically earliest such sequence with no duplicated terms).

Explanation:
The 1st digit of T (2) is the 2nd digit of T (the 2 of 20);
The 2nd digit of T (2) is the 20th digit of T (the 2 of 52);
The 3rd digit of T (0) is the 6th digit of T (the 0 of 10);
The 4th digit of T (6) is the 10th digit of T (the 6 of 60);
The 5th digit of T (1) is the 8th digit of T (the 1 of 11);
The 6th digit of T (0) is the 11th digit of T (the 0 of 60);
The 7th digit of T (8) is the 60th digit of T (8 – not visible here);
The 8th digit of T (1) is the 5th digit of T (the 1 of 10);
Etc.

The yellow column here:

https://bit.ly/2JzRaZW

... is the sequence itself.

Best,
É.