New Sequence Idea

[Dave Consiglio]

Hello all,

What do you think of the following sequence?

a(n) is the smallest positive integer not present as a substring of any term of the first n terms of the Fibonacci sequence:

[2, 2, 3, 4, 4, 4, 4, 4, 6, 6, 6, 6, 6, 6, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 38, 38, 38, 38, 38, 38, 38, 38, 38, 38, 38, 38, 38, 38, 38, 38, 38, 47, 47, 47, 47, 47, 47, 66, 66, 66, 66, 66, 66, 66, 66, 82, 82, 82, 82, 82, 82, 82, 82, 82, 82, 82, 82, 82, 82, 82, 82, 82, 82, 82, 82, 82, 82, 82, 82, 82, 104, 104, 104, 104, 104, 104, 104]

Example:  a(9) = 6 because the first 9 terms of the Fibonacci sequence are [1,1,2,3,5,8,13,21,34] and 1, 2, 3, 4, and 5 all appear as substrings of at least one term in this subset of Fibonacci sequence.

This sequence grows very slowly (a(10000) = 10010). Is it infinite?

If we concatenate the terms of the Fibonacci sequence, we instead get:

[2, 2, 3, 4, 4, 4, 4, 4, 6, 6, 6, 6, 6, 6, 15, 15, 16, 16, 16, 17, 17, 19, 19, 19, 19, 19, 20, 20, 20, 24, 24, 24, 27, 27, 38, 38, 38, 38, 38, 38, 38, 38, 38, 38, 38, 38, 38, 38, 38, 38, 38, 38, 38, 47, 47, 47, 47, 47, 47, 47, 66, 66, 66, 66, 66, 66, 66, 66, 82, 82, 82, 82, 82, 82, 82, 82, 82, 82, 82, 82, 82, 82, 82, 82, 100, 100, 100, 103, 103, 103, 104, 104, 104, 104, 104, 104, 104, 104, 104, 104]

Note that 11 is no longer in the sequence because the concatenated Fibonacci sequence is

112358132134... and '11' is present in the first two digits.

This could also be extended to other sequences. For example, the digits of Pi:

[2, 2, 2, 2, 2, 2, 6, 7, 7, 7, 7, 7, 7, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12]

Do you think sequences of this type merit inclusion in the OEIS? I've searched for several and found nothing like this there.

Thanks,

Dave Consiglio

[David desJardins]

Since you asked ... I think the point of OEIS is to catalog sequences that might actually appear for someone else in some possibly different context.

Not sure what the point of something like this is.

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[Neil Sloane]

Hi Dave,  Since you asked, I have to say that sequences based on the concatenation of the digits of the Fibonacci sequence are in my opinion  not appropriate for the OEIS.

They are artificial, base dependent, and tedious (which is why you didn't find any examples in the OEIS).

Why not read some real mathematics?  Here are two of my favorite books:

Tom Apostol, Into to Analytic Number Theory,

Mark Kac, Statistical Independence in Probability, Analysis and Number Theory,

Tom Körner, The Pleasures of Counting.

Best regards

Neil 

Neil J. A. Sloane, Chairman, OEIS Foundation.

Also Visiting Scientist, Math. Dept., Rutgers University, 

Email: njas...@gmail.com

On Mon, Apr 6, 2026 at 11:46 AM Dave Consiglio <dave...@gmail.com> wrote:

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[Dave Consiglio]

Hi all,

Thanks for the feedback. I'll shelve these. :)

And thanks for the reading suggestions, Neil.

Dave

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[Antti Karttunen]

On Mon, Apr 6, 2026 at 7:20 PM Dave Consiglio <dave...@gmail.com> wrote:

Hi all,

Thanks for the feedback. I'll shelve these. :) 

And thanks for the reading suggestions, Neil.

Dave

On Mon, Apr 6, 2026 at 10:04 AM Neil Sloane <njas...@gmail.com> wrote:

Hi Dave,  Since you asked, I have to say that sequences based on the concatenation of the digits of the Fibonacci sequence are in my opinion  not appropriate for the OEIS.

They are artificial, base dependent, and tedious (which is why you didn't find any examples in the OEIS).

Why not read some real mathematics?  Here are two of my favorite books:

Tom Apostol, Into to Analytic Number Theory,

Mark Kac, Statistical Independence in Probability, Analysis and Number Theory,

Tom Körner, The Pleasures of Counting.

Cheers,

May I suggest also:

Paul Zeitz: The Art and Craft of Problem Solving.

That book gives a good introduction to many concepts in the number theory and the combinatorics, explained in very down to earth fashion. Could be difficult to find an affordable copy, though.

Best,

Antti

 

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[jpallouche.math]

About that last book, I would certainly not write that
a very affordable copy (uh possibly not 100% legal) 
is said by some people to be obtainable on the web.

jp

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