Similar to boustrophedon transform of 1,1,1,1,1,...

[Davide Rotondo]

Hi to all dear seqfans! it's the very first time I noticed in my researches about boustrophedon transform so I want to share with you a similar transform method that I constructed using a little bit complicate Triangle. If something new will emerge I will be very happy! 

What do you think?

Davide

[Davide Rotondo]

[Md. Rad Sarar Anando]

Didn't get it clearly. Please elaborate this framework. 

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[Davide Rotondo]

                                    1          3        1       35         1       1123      1

                               1         2         4       34       36      1122  1124

                         9         8          6       30      70     1086   2246

                    1       10       18       24     100    1016   3332

            177    176    166     148    124    916    4348

         1      178     354    520     668    792    5264

8569  8568  8390  8036  7516  6848  6056

Ok like this?

Davide

[Md. Rad Sarar Anando]

OK, give me time. I need a deep analysis on this. I will let you know after I'm done with it. Thanks for your explanation.

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[Allan Wechsler]

Davide's diagram is a double triangle. In each "row", you start at one end with 1, and repeatedly add the corresponding element of the previous row. At the "kink" in the row, you use the center element of the previous row twice, for example 4+2 = 6, 6+2 = 8 in the third row.

The only other wrinkle is that Davide builds the rows of the diagram in alternating directions, so the end you choose to start with alternates between left and right. (I think if you don't alternate ends, you get the powers of three.) At any rate, the resulting sequence seems to be A000834, so I'm not sure what novelty Davide has contributed.

-- Allan

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[Davide Rotondo]

Dear Allan I thought so too at first but then it changed.

Davide

[Allan Wechsler]

Davide, all the numbers that you actually list (1, 3, 9, 35, 177, 1123, 8569) match A000834. If a later entry disagrees, please show that entry. It would be even better if you could write a program that produces your sequence, because hand calculations are always error-prone.

-- Allan

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[Md. Rad Sarar Anando]

Dear Davide, 
The frame you provided also arises as the endpoint values of a nonstandard cumulative-sum like “double triangle,” where each row is formed by adding corresponding entries of the previous row, the central entry is used twice at a kink, and rows are built boustrophedonically (alternating left/right). If the alternation is suppressed, the same construction yields powers of 3, suggesting that the directional alternation alone induces the transition to A000834. Anyway, after my close attention, I got that your idea already exists in OEIS  A000834 but your clarification was a bit surprising. Anyway, your thoughts about this should be added to "Comment" as a new insight if you wish. My observation says something like this, " The interest lies in isolating directional alternation as the sole source of complexity: without alternation the same rule yields powers of 3, while with alternation it produces A000834."

Rigards, 
Rad

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[Davide Rotondo]

Thanks a lot Rad!

Davide

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