[seqfan] Re: 0-additive arrays

[Lars Blomberg]

Hello,

I find the following for numbers up to 10000.

Sequences
1) 1, 2, 4, 7, 10, 13, 16, 19, 22, 25, 28, 31, 34, 37, 40, 43, 46, 49, 52, 55, 58, 61, 64, 67, 70, 73, 76, 79, 82, 85, 88, 91, 94, 97, 100, 103, 106, 109
2) 3, 5, 6, 12, 14, 21, 23, 30, 32, 39, 41, 48, 50, 57, 59, 66, 68, 75, 77, 84, 86, 93, 95, 102, 104, 111, 113, 120, 122, 129, 131, 138, 140, 147, 149
3) 8, 9, 11, 15, 18, 35, 36, 38, 42, 62, 63, 65, 69, 89, 90, 92, 96, 116, 117, 119, 123, 143, 144, 146, 150, 170, 171, 173, 177, 197, 198, 200, 204, 224
4) 17, 20, 24, 26, 27, 29, 33, 45, 54, 98, 101, 105, 107, 108, 110, 114, 126, 179, 182, 186, 188, 189, 191, 195, 207, 260, 263, 267, 269, 270, 272, 276
5) 44, 47, 51, 53, 56, 60, 71, 72, 74, 78, 80, 81, 83, 87, 99, 135, 162, 287, 290, 294, 296, 299, 303, 314, 315, 317, 321, 323, 324, 326, 330, 342, 378
6) 125, 128, 132, 134, 137, 141, 152, 153, 155, 159, 161, 164, 168, 180, 206, 209, 213, 215, 216, 218, 222, 233, 234, 236, 240, 242, 243, 245, 249, 261
7) 368, 371, 375, 377, 380, 384, 395, 396, 398, 402, 404, 407, 411, 423, 449, 452, 456, 458, 459, 461, 465, 476, 477, 479, 483, 485, 488, 492, 504, 540

The first term of the sequences (Not in OEIS)
1,3,8,17,44,125,368

Differences
1) 1, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3
2) 2, 1, 6, 2, 7, 2, 7, 2, 7, 2, 7, 2, 7, 2, 7, 2, 7, 2, 7, 2, 7, 2, 7, 2, 7, 2, 7, 2, 7, 2, 7, 2, 7, 2, 7, 2, 7, 2, 7, 2, 7, 2, 7, 2, 7, 2, 7, 2, 7, 2
3) 1, 2, 4, 3, 17, 1, 2, 4, 20, 1, 2, 4, 20, 1, 2, 4, 20, 1, 2, 4, 20, 1, 2, 4, 20, 1, 2, 4, 20, 1, 2, 4, 20, 1, 2, 4, 20, 1, 2, 4, 20, 1, 2, 4, 20, 1, 2
4) 3, 4, 2, 1, 2, 4, 12, 9, 44, 3, 4, 2, 1, 2, 4, 12, 53, 3, 4, 2, 1, 2, 4, 12, 53, 3, 4, 2, 1, 2, 4, 12, 53, 3, 4, 2, 1, 2, 4, 12, 53, 3, 4, 2, 1, 2, 4
5) 3, 4, 2, 3, 4, 11, 1, 2, 4, 2, 1, 2, 4, 12, 36, 27, 125, 3, 4, 2, 3, 4, 11, 1, 2, 4, 2, 1, 2, 4, 12, 36, 152, 3, 4, 2, 3, 4, 11, 1, 2, 4, 2, 1, 2, 4
6) 3, 4, 2, 3, 4, 11, 1, 2, 4, 2, 3, 4, 12, 26, 3, 4, 2, 1, 2, 4, 11, 1, 2, 4, 2, 1, 2, 4, 12, 36, 108, 81, 368, 3, 4, 2, 3, 4, 11, 1, 2, 4, 2, 3, 4, 12
7) 3, 4, 2, 3, 4, 11, 1, 2, 4, 2, 3, 4, 12, 26, 3, 4, 2, 1, 2, 4, 11, 1, 2, 4, 2, 3, 4, 12, 36, 71, 3, 4, 2, 3, 4, 11, 1, 2, 4, 2, 1, 2, 4, 12, 26, 3, 4
The longer ones have a common start.

As Eric noted, there are cycles in the differences:

Difference cycles
1) 3
2) 2, 7
3) 1, 2, 4, 20
4) 3, 4, 2, 1, 2, 4, 12, 53
5) 3, 4, 2, 3, 4, 11, 1, 2, 4, 2, 1, 2, 4, 12, 36, 152
6) 3, 4, 2, 3, 4, 11, 1, 2, 4, 2, 3, 4, 12, 26, 3, 4, 2, 1, 2, 4, 11, 1, 2, 4, 2, 1, 2, 4, 12, 36, 108, 449
7) 3, 4, 2, 3, 4, 11, 1, 2, 4, 2, 3, 4, 12, 26, 3, 4, 2, 1, 2, 4, 11, 1, 2, 4, 2, 3, 4, 12, 36, 71, 3, 4, 2, 3, 4, 11, 1, 2, 4, 2, 1, 2, 4, 12, 26, 3, 4, 2, 1, 2, 4, 11, 1, 2, 4, 2, 1, 2, 4, 12, 36, 108, 324, 1340
The cycles have several subsequences in common.

Length of the difference cycles (A000079 = 2^n)
1, 2, 4, 8, 16, 32, 64, 128

Sum of the difference cycles (A000244 = Powers of 3)
3, 9, 27, 81, 243, 729, 2187, 6561

Furthermore, the differences starts with almost a difference cycle,
only that the last term is split into 2.
For example 3) has cycle
1, 2, 4, 20
and the differences start
1, 2, 4, 3, 17, and then 1, 2, 4, 20 is repeated.

Therefore we have:
Index for first start of the difference cycle (A000051 = 2^n + 1)
2, 3, 5, 9, 17, 33, 65

The first part of the difference cycle split (A000244 = Powers of 3, except for first term)
1, 1, 3, 9, 27, 81, 243, 729

The second part of the difference cycle split (Not in OEIS)
2, 6, 17, 44, 125, 368, 1097, 3284

The largest value in the difference cycle (Not in OEIS)
3, 7, 20, 53, 152, 449, 1340

The next-to-largest value in the difference cycle (A025579 = a(1)=1, a(2)=2, a(n) = 4*3^(n-3) for n >= 3)
2, 4, 12, 36, 108, 324

In fact, the cycles end with the a few of the initial values of
1, 2, 4, 12, 36, 108, 324
followed by the last (and largest) value.

/Lars

-----Ursprungligt meddelande-----
Från: SeqFan [mailto:seqfan-bounces at list.seqfan.eu] För Olivier Gerard
Skickat: den 16 maj 2015 20:16
Till: Sequence Fanatics Discussion list
Ämne: [seqfan] Re: 0-additive arrays

On Sat, May 16, 2015 at 3:56 PM, Eric Angelini <Eric.Angelini at kntv.be>
wrote:

>
> Hello SeqFans,
> Here is a way to build an infinite 0-additive array, I guess.
>
> Always try to fill the leftmost empty cell of the top row with the 
> smallest unused integer "a" so far.
>
> If this is not possible, try to fill the leftmost empty cell of the 
> next row with "a".
>
> If this is not possible, try to fill the leftmost empty cell of the 
> next row with "a" (etc.)
>
> Reminder: 0-additivity means that no "a" is the sum of any two 
> integers belonging to the row where "a" stands.
>
>
> [...]

>
> [Sorry for the long post, Gérard -- especially if this is old hat.
>

For variants of the same idea,
you might want to lookup  Interpersion array, Wythoff array, Stolarsky array and speak with Clark Kimberling from Evansville University.


> [Gérard is the list-admin, as you all know]
>

Gérard is my family name. You can call me Olivier.

Regards,

Olivier Gérard

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