# [seqfan] Re: reply to Message 4. Anyone recognize this matrix? (Neil Sloane) (Marc LeBrun) (I. V. Serov)

I.V. Serov i.v.serov at chf.nu
Wed Jan 5 12:11:20 CET 2022

```Dear SeqFans, Marc, Neil,

Please, fill free to publish the sequence and I am glad to see that it
is proving to be promising on New Year's Day!

Best regards,
I. V. Serov
www.chf.nu

On 01-01-2022 20:48, Marc LeBrun wrote:
> Just responding to an old thread (no idea if this was subsequently
>
> I would suggest that rather than rearranging an existing sequence,
> that a new sequence be added, with the new formula -- then both
> sequences outfitted with appropriate cross-references to each other
> and to A341915 and A341916, etc.
>
>
>
>> On Apr 22, 2021, at 1:25 PM, I.V. Serov <i.v.serov at chf.nu> wrote:
>>
>> Dear Neil,
>>
>> Would you allow to rearrange the order of the rows in the matrix?
>>
>> It looks like the same rows can be generated (in a different order) by
>> means of a formula:
>>
>> B(1) = 01.
>> B(2n+0) = concatenate(B'(n);B(n)).
>> B(2n+1) = concatenate(B(n);B(n)').
>> Here B(n)' equals B(n) with the rightmost bit flipped.
>>
>> These are the first 31 rows of your matrix in the new order:
>>
>> 01*
>> 0001*
>> 0100*
>> 00000001*
>> 00010000*
>> 01010100*
>> 01000101*
>> 0000000000000001*
>> 0000000100000000*
>> 0001000100010000*
>> 0001000000010001*
>> 0101010101010100*
>> 0101010001010101*
>> 0100010001000101*
>> 0100010101000100*
>> 00000000000000000000000000000001*
>> 00000000000000010000000000000000*
>> 00000001000000010000000100000000*
>> 00000001000000000000000100000001*
>> 00010001000100010001000100010000*
>> 00010001000100000001000100010001*
>> 00010000000100000001000000010001*
>> 00010000000100010001000000010000*
>> 01010101010101010101010101010100*
>> 01010101010101000101010101010101*
>> 01010100010101000101010001010101*
>> 01010100010101010101010001010100*
>> 01000100010001000100010001000101*
>> 01000100010001010100010001000100*
>> 01000101010001010100010101000100*
>> 01000101010001000100010101000101*
>>
>> The order permutations are described by A341915 and A341916.
>>
>>
>> Kind regards,
>>
>> Igor Serov
>> www.chf.nu
>>
>>
>>
>> On 21-04-2021 15:07, seqfan-request at list.seqfan.eu wrote:
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>>>   1. A339950, A189378 (Don Reble)
>>>   2. Re: A339950, A189378 (Jeffrey Shallit)
>>>   3. How to define analogue to this sequence pertaining to Roman
>>>      numerals (Alonso Del Arte)
>>>   4. Anyone recognize this matrix? (Neil Sloane)
>>>   5. Recent perturbations on the Seqfan Mailing List (Olivier Gerard)
>>>   6. Conjecture: a(n) = n only if n = 1 or 9 (Alonso Del Arte)
>>>   7. Lucasian (pseudo)primes (Tomasz Ordowski)
>>>   8. Re: Conjecture: a(n) = n only if n = 1 or 9 (D. S. McNeil)
>>>   9. Sum-Product Problem (Frank Adams-watters)
>>>  10. Planar distributive lattices (Allan Wechsler)
>>>  11. Re: Sum-Product Problem (Neil Sloane)
>>>  12. Re: Sum-Product Problem (Neil Sloane)
>>>  13. Re: Planar distributive lattices (Neil Sloane)
>>>  14. Re: Planar distributive lattices (Neil Sloane)
>>>  15. Re: Sum-Product Problem (Hugo Pfoertner)
>>>  16. Re: Conjecture: a(n) = n only if n = 1 or 9 (Alonso Del Arte)
>>>  17. Re: Planar distributive lattices (Allan Wechsler)
>>>  18. Re: Planar distributive lattices (Neil Sloane)
>>>  19. help naming/describing sequences for bounds of Goldbach's
>>>      Comet, was: what happened to my A342302, its been replaced ???
>>>      (peter lawrence)
>>>  20. Re: Anyone recognize this matrix? (Richard J. Mathar)
>>>  21. Re: Anyone recognize this matrix? (Neil Sloane)
>>> ----------------------------------------------------------------------
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>> ------------------------------
>>> Message: 4
>>> Date: Fri, 9 Apr 2021 22:06:31 -0400
>>> From: Neil Sloane <njasloane at gmail.com>
>>> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
>>> Subject: [seqfan] Anyone recognize this matrix?
>>> Message-ID:
>>> 	<CAAOnSgS89HNuj2uW5NMGx_qCPV8mu_h_Gi=wLJR-eROGQXhzVA at mail.gmail.com>
>>> Content-Type: text/plain; charset="UTF-8"
>>> Dear Sequence Fans, I have an infinite 0,1 matrix. The first row is
>>> 01
>>> repeated, the second row is 0100 repeated, and so on. Here are the
>>> first 32
>>> rows.
>>> I have a feeling I've seen this before, but I can't remember where.
>>> I have the definition, but I would like a simple description.
>>> Does anyone recognize this?
>>> There are some obvious properties. In rows 8 through 15, for
>>> instance, the
>>> mod 2 sums row 8 + row 15 = row 9 + row 14 = ... = row 11 + row 12 =
>>> 0000000100000001.
>>> And similarly for rows 2 to 3; 4 to 7; 16 to 31; etc.
>>> 1: 01*
>>> 2: 0100*
>>> 3: 0001*
>>> 4: 00010000*
>>> 5: 01000101*
>>> 6: 01010100*
>>> 7: 00000001*
>>> 8: 0000000100000000*
>>> 9: 0101010001010101*
>>> 10: 0100010101000100*
>>> 11: 0001000000010001*
>>> 12: 0001000100010000*
>>> 13: 0100010001000101*
>>> 14: 0101010101010100*
>>> 15: 0000000000000001*
>>> 16: 00000000000000010000000000000000*
>>> 17: 01010101010101000101010101010101*
>>> 18: 01000100010001010100010001000100*
>>> 19: 00010001000100000001000100010001*
>>> 20: 00010000000100010001000000010000*
>>> 21: 01000101010001000100010101000101*
>>> 22: 01010100010101010101010001010100*
>>> 23: 00000001000000000000000100000001*
>>> 24: 00000001000000010000000100000000*
>>> 25: 01010100010101000101010001010101*
>>> 26: 01000101010001010100010101000100*
>>> 27: 00010000000100000001000000010001*
>>> 28: 00010001000100010001000100010000*
>>> 29: 01000100010001000100010001000101*
>>> 30: 01010101010101010101010101010100*
>>> 31: 00000000000000000000000000000001*
>>> [These are actually the odd-numbered rows 1,3,5,7,... of the matrix.
>>> The
>>> even-numbered rows have a simple formula. Row 2k is 0^(2^m) 1^(2^m)
>>> repeated, where m is the number of times 2 divides 2k.
>>> Row 24 for example (where m=3) is 0000000011111111 repeated. I'm
>>> hoping for
>>> something similar for the odd-numbered rows.]
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>> ------------------------------
>>> Message: 20
>>> Date: Wed, 21 Apr 2021 14:47:35 +0200
>>> From: "Richard J. Mathar" <mathar at mpia-hd.mpg.de>
>>> To: seqfan at list.seqfan.eu
>>> Subject: [seqfan] Re: Anyone recognize this matrix?
>>> Message-ID: <20210421124735.GA20127 at mathar.mpia-hd.mpg.de>
>>> Content-Type: text/plain; charset=us-ascii
>>> A formal description of this infinite array of 0's and 1's is:
>>> The "full" array including a leading row of all-0 starts as follows:
>>> 0 00000000000000000
>>> 1 01010101010101010
>>> 2 01000100010001000
>>> 3 00010001000100010
>>> 4 00010000000100000
>>> 5 01000101010001010
>>> 6 01010100010101000
>>> 7 00000001000000010
>>> 8 00000001000000000
>>> 9 01010100010101010
>>> 10 01000101010001000
>>> 11 00010000000100010
>>> 12 00010001000100000
>>> 13 01000100010001010
>>> 14 01010101010101000
>>> 15 00000000000000010
>>> 16 00000000000000010
>>> 17 01010101010101000
>>> 18 01000100010001010
>>> 19 00010001000100000
>>> 20 00010000000100010
>>> 21 01000101010001000
>>> 22 01010100010101010
>>> 23 00000001000000000
>>> 24 00000001000000010
>>> 25 01010100010101000
>>> 26 01000101010001010
>>> 27 00010000000100000
>>> 28 00010001000100010
>>> 29 01000100010001000
>>> 30 01010101010101010
>>> 31 00000000000000000
>>> Because each second column contains only zeros, we delete each second
>>> column
>>> and get the "reduced" array
>>> 0 00000000000000000
>>> 1 11111111111111111
>>> 2 10101010101010101
>>> 3 01010101010101010
>>> 4 01000100010001000
>>> 5 10111011101110111
>>> 6 11101110111011101
>>> 7 00010001000100010
>>> 8 00010000000100000
>>> 9 11101111111011111
>>> 10 10111010101110101
>>> 11 01000101010001010
>>> 12 01010100010101000
>>> 13 10101011101010111
>>> 14 11111110111111101
>>> 15 00000001000000010
>>> 16 00000001000000000
>>> 17 11111110111111111
>>> 18 10101011101010101
>>> 19 01010100010101010
>>> 20 01000101010001000
>>> 21 10111010101110111
>>> 22 11101111111011101
>>> 23 00010000000100010
>>> 24 00010001000100000
>>> 25 11101110111011111
>>> 26 10111011101110101
>>> 27 01000100010001010
>>> 28 01010101010101000
>>> 29 10101010101010111
>>> 30 11111111111111101
>>> 31 00000000000000010
>>> Each odd-numbered row is the binary complement of its preceding row,
>>> so
>>> define a "depleted reduced" array just containing rows 0,2,4,6,8,...:
>>> 0 00000000000000000
>>> 2 10101010101010101
>>> 4 01000100010001000
>>> 6 11101110111011101
>>> 8 00010000000100000
>>> 10 10111010101110101
>>> 12 01010100010101000
>>> 14 11111110111111101
>>> 16 00000001000000000
>>> 18 10101011101010101
>>> 20 01000101010001000
>>> 22 11101111111011101
>>> 24 00010001000100000
>>> 26 10111011101110101
>>> 28 01010101010101000
>>> 30 11111111111111101
>>> The definition of this seems to be given by reading the 1st, 2nd, 3rd
>>> ... column downwards, which gives periodic patterns of zeros and
>>> ones:
>>> 0,1 (col 1)
>>> 0,0,1,1 (col 2)
>>> 0,1 (col 3)
>>> 0,0,0,0,1,1,1,1 (col 4)
>>> 0,1 (col 5)
>>> 0,0,1,1 (col 6)
>>> 0,1 (col 7)
>>> 0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1 (col 8)
>>> 0,1 (col 9)
>>> 0,0,1,1 (col 10)
>>> 0,1 (col 11)
>>> 0,0,0,0,1,1,1,1 (col 12)
>>> 0,1 (col 13)
>>> 0,0,1,1 (col 14)
>>> 0,1 (col 15)
>>> 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1, (col
>>> 16)
>>> where the number of zeros (and number of ones) in the periods
>>> of column k is given by A006519(k).
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>> ------------------------------
>>> Message: 21
>>> Date: Wed, 21 Apr 2021 09:07:21 -0400
>>> From: Neil Sloane <njasloane at gmail.com>
>>> To: Sequence Fanatics Discussion list <seqfan at list.seqfan.eu>
>>> Subject: [seqfan] Re: Anyone recognize this matrix?
>>> Message-ID:
>>> 	<CAAOnSgR1XuMEHXrhCRSXK7ujKXfTMMjzyZUKQL1_gm3SwDo__Q at mail.gmail.com>
>>> Content-Type: text/plain; charset="UTF-8"
>>> Richard,  You are right, and indeed one can say much more. In fact
>>> this is
>>> part of a bigger investigation and there is a paper in progress that
>>> will
>>> reveal everything. I was hoping to have it finished a week ago, but
>>> keeping
>>> the OEIS running takes a great deal of time.  Once the paper is in
>>> form I will post a link to it here.
>>> Best regards
>>> Neil
>>> Neil J. A. Sloane, President, OEIS Foundation.
>>> 11 South Adelaide Avenue, Highland Park, NJ 08904, USA.
>>> Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway,
>>> NJ.
>>> Phone: 732 828 6098; home page: http://NeilSloane.com
>>> Email: njasloane at gmail.com
>>> On Wed, Apr 21, 2021 at 8:47 AM Richard J. Mathar
>>> <mathar at mpia-hd.mpg.de>
>>> wrote:
>>>> A formal description of this infinite array of 0's and 1's is:
>>>> The "full" array including a leading row of all-0 starts as follows:
>>>> 0 00000000000000000
>>>> 1 01010101010101010
>>>> 2 01000100010001000
>>>> 3 00010001000100010
>>>> 4 00010000000100000
>>>> 5 01000101010001010
>>>> 6 01010100010101000
>>>> 7 00000001000000010
>>>> 8 00000001000000000
>>>> 9 01010100010101010
>>>> 10 01000101010001000
>>>> 11 00010000000100010
>>>> 12 00010001000100000
>>>> 13 01000100010001010
>>>> 14 01010101010101000
>>>> 15 00000000000000010
>>>> 16 00000000000000010
>>>> 17 01010101010101000
>>>> 18 01000100010001010
>>>> 19 00010001000100000
>>>> 20 00010000000100010
>>>> 21 01000101010001000
>>>> 22 01010100010101010
>>>> 23 00000001000000000
>>>> 24 00000001000000010
>>>> 25 01010100010101000
>>>> 26 01000101010001010
>>>> 27 00010000000100000
>>>> 28 00010001000100010
>>>> 29 01000100010001000
>>>> 30 01010101010101010
>>>> 31 00000000000000000
>>>> Because each second column contains only zeros, we delete each
>>>> second
>>>> column
>>>> and get the "reduced" array
>>>> 0 00000000000000000
>>>> 1 11111111111111111
>>>> 2 10101010101010101
>>>> 3 01010101010101010
>>>> 4 01000100010001000
>>>> 5 10111011101110111
>>>> 6 11101110111011101
>>>> 7 00010001000100010
>>>> 8 00010000000100000
>>>> 9 11101111111011111
>>>> 10 10111010101110101
>>>> 11 01000101010001010
>>>> 12 01010100010101000
>>>> 13 10101011101010111
>>>> 14 11111110111111101
>>>> 15 00000001000000010
>>>> 16 00000001000000000
>>>> 17 11111110111111111
>>>> 18 10101011101010101
>>>> 19 01010100010101010
>>>> 20 01000101010001000
>>>> 21 10111010101110111
>>>> 22 11101111111011101
>>>> 23 00010000000100010
>>>> 24 00010001000100000
>>>> 25 11101110111011111
>>>> 26 10111011101110101
>>>> 27 01000100010001010
>>>> 28 01010101010101000
>>>> 29 10101010101010111
>>>> 30 11111111111111101
>>>> 31 00000000000000010
>>>> Each odd-numbered row is the binary complement of its preceding row,
>>>> so
>>>> define a "depleted reduced" array just containing rows
>>>> 0,2,4,6,8,...:
>>>> 0 00000000000000000
>>>> 2 10101010101010101
>>>> 4 01000100010001000
>>>> 6 11101110111011101
>>>> 8 00010000000100000
>>>> 10 10111010101110101
>>>> 12 01010100010101000
>>>> 14 11111110111111101
>>>> 16 00000001000000000
>>>> 18 10101011101010101
>>>> 20 01000101010001000
>>>> 22 11101111111011101
>>>> 24 00010001000100000
>>>> 26 10111011101110101
>>>> 28 01010101010101000
>>>> 30 11111111111111101
>>>> The definition of this seems to be given by reading the 1st, 2nd,
>>>> 3rd
>>>> ... column downwards, which gives periodic patterns of zeros and
>>>> ones:
>>>> 0,1 (col 1)
>>>> 0,0,1,1 (col 2)
>>>> 0,1 (col 3)
>>>> 0,0,0,0,1,1,1,1 (col 4)
>>>> 0,1 (col 5)
>>>> 0,0,1,1 (col 6)
>>>> 0,1 (col 7)
>>>> 0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1 (col 8)
>>>> 0,1 (col 9)
>>>> 0,0,1,1 (col 10)
>>>> 0,1 (col 11)
>>>> 0,0,0,0,1,1,1,1 (col 12)
>>>> 0,1 (col 13)
>>>> 0,0,1,1 (col 14)
>>>> 0,1 (col 15)
>>>> 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
>>>> (col 16)
>>>> where the number of zeros (and number of ones) in the periods
>>>> of column k is given by A006519(k).
>>>> --
>>>> Seqfan Mailing list - http://list.seqfan.eu/
>>> ------------------------------
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>>
>>> _______________________________________________
>>> SeqFan mailing list
>>> SeqFan at list.seqfan.eu
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>>> ------------------------------
>>> End of SeqFan Digest, Vol 151, Issue 1
>>> End of SeqFan Digest, Vol 159, Issue 1
>>> **************************************
>>
>> --
>>
>> --
>> Seqfan Mailing list - http://list.seqfan.eu/

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