Two self-describing sequences for SeqFans

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 5^th letter of the sequence:

F

Go on with the 14^th:

FI

And the 84^th:

FIV

The 10^th:

FIVE

The 1^st:

FIVE F

The 20^th:

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 playing with this idea is of no interest:

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

We thus decided that a term of S cannot command the duplication of one of its own letters – the letter to be duplicated must be found elsewhere, up or downstream [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)].

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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 1^st digit of T (2) is the 2^nd digit of T (the 2 of 20);

The 2^nd digit of T (2) is the 20^th digit of T (the 2 of 52);

The 3^rd digit of T (0) is the 6^th digit of T (the 0 of 10);

The 4^th digit of T (6) is the 10^th digit of T (the 6 of 60);

The 5^th digit of T (1) is the 8^th digit of T (the 1 of 11);

The 6^th digit of T (0) is the 11^th digit of T (the 0 of 60);

The 7^th digit of T (8) is the 60^th digit of T (8 – not visible here);

The 8^th digit of T (1) is the 5^th digit of T (the 1 of 10);

Etc.

(The yellow column is the sequence itself).

Best,

É.

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March 23rd 2019, close to midnight, Brussels (Belgium) time.